**The proof of Lemma 1449 (Lagrange polynomials is basis of $\mathbb{P}_k^1$)**

cites explicitly:

Lemma 1376 (free family of $\dim$ elements is basis),

Definition 1417 (polynomial space $\mathbb{P}_k^1$),

Lemma 1418 ($\mathbb{P}_k^1$ is space of degree at most $k$),

Lemma 1420 (dimension of $\mathbb{P}_k^1$),

Lemma 1422 (product of two univariate polynomials),

Definition 1448 (Lagrange polynomials of $\mathbb{P}_k^1$).

**The proof of Lemma 1450 (decomposition of $\mathbb{P}_k^1$ polynomial in Lagrange basis)**

cites explicitly:

Lemma 1449 (Lagrange polynomials is basis of $\mathbb{P}_k^1$).

**The proof of Lemma 1452 (reference simplex is nontrivial in $\mathbb{R}$)**

cites explicitly:

Definition 1437 (reference simplex),

Lemma 1439 (nontrivial reference simplex).

**The proof of Lemma 1454 (reference Lagrange nodes are distinct)**

cites explicitly:

Definition 1453 (reference Lagrange nodes of $\mathbb{P}_k^1$).

**The proof of Lemma 1455 (reference Lagrange basis of $\mathbb{P}_k^1$)**

cites explicitly:

Lemma 1449 (Lagrange polynomials is basis of $\mathbb{P}_k^1$),

Lemma 1454 (reference Lagrange nodes are distinct).

**The proof of Lemma 1457 (reference Lagrange linear forms for $\mathbb{P}_k^1$ are linear)**

cites explicitly:

Definition 1456 (reference Lagrange linear forms for $\mathbb{P}_k^1$).

**The proof of Lemma 1458 (reference Lagrange linear forms for $\mathbb{P}_k^1$ are injective)**

cites explicitly:

Statement(s) from [14],

Definition 1424 (finite element triple),

Lemma 1450 (decomposition of $\mathbb{P}_k^1$ polynomial in Lagrange basis),

Lemma 1454 (reference Lagrange nodes are distinct),

Definition 1456 (reference Lagrange linear forms for $\mathbb{P}_k^1$).

**The proof of Lemma 1459 (unisolvence of $\mathbb{P}_k^1$ (reference))**

cites explicitly:

Lemma 1420 (dimension of $\mathbb{P}_k^1$),

Lemma 1426 (injectivity implies unisolvence),

Lemma 1454 (reference Lagrange nodes are distinct),

Lemma 1458 (reference Lagrange linear forms for $\mathbb{P}_k^1$ are injective).

**The proof of Theorem 1460 (Log $\hat{\mathbb{P}}_k^1$ reference Lagrange finite element)**

cites explicitly:

Lemma 1420 (dimension of $\mathbb{P}_k^1$),

Definition 1424 (finite element triple),

Lemma 1452 (reference simplex is nontrivial in $\mathbb{R}$),

Lemma 1459 (unisolvence of $\mathbb{P}_k^1$ (reference)).

**The proof of Lemma 1462 (properties of geometric mapping in dimension 1)**

cites explicitly:

Lemma 1399 (equivalent definition of affine map (finite dimension)),

Lemma 1410 (inverse of affine map is affine map),
Lemma 1418 ($\mathbb{P}_k^1$ is space of degree at most $k$),

Definition 1453 (reference Lagrange nodes of $\mathbb{P}_k^1$),

Lemma 1455 (reference Lagrange basis of $\mathbb{P}_k^1$),

Definition 1461 (geometric mapping in dimension 1).

**The proof of Lemma 1463 (current simplex is nontrivial in $\mathbb{R}$)**

cites explicitly:

Definition 1440 (simplex).

**The proof of Lemma 1464 (current simplex is image of reference in $\mathbb{R}$)**

cites explicitly:

Lemma 1452 (reference simplex is nontrivial in $\mathbb{R}$),

Definition 1461 (geometric mapping in dimension 1),

Lemma 1463 (current simplex is nontrivial in $\mathbb{R}$).

**The proof of Lemma 1467 (Lagrange nodes are distinct)**

cites explicitly:

Definition 1465 (Lagrange nodes of $\mathbb{P}_k^1$).

**The proof of Lemma 1468 (Lagrange nodes of $\mathbb{P}_k^1$ are images of reference)**

cites explicitly:

Definition 1453 (reference Lagrange nodes of $\mathbb{P}_k^1$),

Definition 1461 (geometric mapping in dimension 1),

Definition 1465 (Lagrange nodes of $\mathbb{P}_k^1$).

**The proof of Lemma 1469 (Lagrange basis of $\mathbb{P}_k^1$)**

cites explicitly:

Lemma 1449 (Lagrange polynomials is basis of $\mathbb{P}_k^1$),

Lemma 1467 (Lagrange nodes are distinct).

**The proof of Lemma 1470 (Lagrange polynomials of $\mathbb{P}_k^1$ are images of reference)**

cites explicitly:

Definition 1448 (Lagrange polynomials of $\mathbb{P}_k^1$),

Lemma 1455 (reference Lagrange basis of $\mathbb{P}_k^1$),

Definition 1461 (geometric mapping in dimension 1),

Lemma 1468 (Lagrange nodes of $\mathbb{P}_k^1$ are images of reference),

Lemma 1469 (Lagrange basis of $\mathbb{P}_k^1$).

**The proof of Lemma 1471 (geometric mapping of $\mathbb{P}_k^1$ is $\mathbb{P}_k^1$)**

cites explicitly:

Definition 1417 (polynomial space $\mathbb{P}_k^1$),

Lemma 1455 (reference Lagrange basis of $\mathbb{P}_k^1$),

Definition 1461 (geometric mapping in dimension 1),

Lemma 1469 (Lagrange basis of $\mathbb{P}_k^1$),

Lemma 1470 (Lagrange polynomials of $\mathbb{P}_k^1$ are images of reference).

**The proof of Lemma 1473 ($\mathbb{P}_k^1$ Lagrange linear forms are linear)**

cites explicitly:

Definition 1472 (Lagrange linear forms for $\mathbb{P}_k^1$).

**The proof of Lemma 1474 ($\mathbb{P}_k^1$ Lagrange linear forms are images of reference)**

cites explicitly:

Definition 1456 (reference Lagrange linear forms for $\mathbb{P}_k^1$),

Lemma 1468 (Lagrange nodes of $\mathbb{P}_k^1$ are images of reference),

Definition 1472 (Lagrange linear forms for $\mathbb{P}_k^1$).
**The proof of Lemma 1476 (Lagrange linear forms for $\mathbb{P}_k^1$ are injective)**

cites explicitly:

Statement(s) from [14],

Definition 1424 (finite element triple),

Lemma 1450 (decomposition of $\mathbb{P}_k^1$ polynomial in Lagrange basis),

Lemma 1467 (Lagrange nodes are distinct),

Definition 1472 (Lagrange linear forms for $\mathbb{P}_k^1$).

**The proof of Lemma 1477 (unisolvence of $\mathbb{P}_k^1$)**

cites explicitly:

Lemma 1420 (dimension of $\mathbb{P}_k^1$),

Lemma 1426 (injectivity implies unisolvence),

Lemma 1467 (Lagrange nodes are distinct),

Lemma 1476 (Lagrange linear forms for $\mathbb{P}_k^1$ are injective).

**The proof of Theorem 1478 ($\text{Log}\mathbb{P}_k^1$ Lagrange finite element)**

cites explicitly:

Lemma 1420 (dimension of $\mathbb{P}_k^1$),

Definition 1424 (finite element triple),

Lemma 1463 (current simplex is nontrivial in $\mathbb{R}$),

Lemma 1477 (unisolvence of $\mathbb{P}_k^1$).

**The proof of Lemma 1481 (length of multi-indices is additive)**

cites explicitly:

Definition 1480 (length of multi-indices).

**The proof of Lemma 1483 (factorial of multi-index is positive)**

cites explicitly:

Definition 1482 (factorial of multi-indices).

**The proof of Lemma 1485 (value of Kronecker delta of multi-indices)**

cites explicitly:

Definition 1484 (Kronecker delta of multi-indices).

**The proof of Lemma 1487 (multi-indices $A_k^d$ for $d=1$ is $A_k^1$)**

cites explicitly:

Definition 1445 (multi-indices $A_k^1$),

Definition 1480 (length of multi-indices),

Definition 1486 (sets of multi-indices $A_k^d$ and $C_k^d$).

**The proof of Lemma 1488 (indices are smaller than maximal length)**

cites explicitly:

Definition 1480 (length of multi-indices),

Definition 1486 (sets of multi-indices $A_k^d$ and $C_k^d$).

**The proof of Lemma 1489 (first $C_k^d$)**

cites explicitly:

Definition 1365 (canonic families),

Definition 1486 (sets of multi-indices $A_k^d$ and $C_k^d$),

Lemma 1488 (indices are smaller than maximal length).

**The proof of Lemma 1493 (slices of $C_k^d$)**

cites explicitly:

Definition 1486 (sets of multi-indices $A_k^d$ and $C_k^d$),

Lemma 1488 (indices are smaller than maximal length),

Definition 1491 (slices $\tilde{S}_{k,i}^d$ and $\tilde{S}_{k,i}^d$).
**The proof of Lemma 1494 (cardinal of slices of $C_k^d$)**

cites explicitly:

Definition 1491 (slices $\tilde{S}_{k,i}^d$ and $\tilde{S}_{k,i}^d$).

**The proof of Lemma 1495 (cardinal of $C_k^d$)**

cites explicitly:

Definition 1362 (binomial coefficient),

Lemma 1364 (properties of the binomial coefficient),

Lemma 1489 (first $C_k^d$),

Lemma 1493 (slices of $C_k^d$),

Lemma 1494 (cardinal of slices of $C_k^d$).

**The proof of Lemma 1496 ($C_k^d$ are layers of $A_k^d$)**

cites explicitly:

Definition 1486 (sets of multi-indices $A_k^d$ and $C_k^d$).

**The proof of Lemma 1497 (first multi-indices $A_k^d$)**

cites explicitly:

Lemma 1489 (first $C_k^d$),

Lemma 1496 ($C_k^d$ are layers of $A_k^d$).

**The proof of Lemma 1498 (cardinal of $A_k^d$)**

cites explicitly:

Lemma 1364 (properties of the binomial coefficient),

Lemma 1495 (cardinal of $C_k^d$),

Lemma 1496 ($C_k^d$ are layers of $A_k^d$).

**The proof of Lemma 1500 (cardinal of $C_k^d$ and $A_k^{d-1}$)**

cites explicitly:

Definition 1480 (length of multi-indices),

Definition 1486 (sets of multi-indices $A_k^d$ and $C_k^d$),

Lemma 1495 (cardinal of $C_k^d$),

Lemma 1498 (cardinal of $A_k^d$).

**The proof of Lemma 1501 (cardinal of $A_{k,i}^d$ and $A_k^{d-1}$)**

cites explicitly:

Definition 1480 (length of multi-indices),

Definition 1486 (sets of multi-indices $A_k^d$ and $C_k^d$).

**The proof of Lemma 1504 (monomial in $d$ variables for $d=1$ is monomial of a single variable)**

cites explicitly:

Definition 1416 (monomial of a single variable),

Lemma 1489 (first $C_k^d$),

Definition 1503 (monomial in $d$ variables).

**The proof of Lemma 1506 (polynomial space $\mathbb{P}_k^d$ for $d=1$ is $\mathbb{P}_k^1$)**

cites explicitly:

Definition 1417 (polynomial space $\mathbb{P}_k^1$),

Lemma 1497 (first multi-indices $A_k^d$),

Definition 1505 (polynomial space $\mathbb{P}_k^d$).

**The proof of Lemma 1507 ($\mathbb{P}_k^d$ is vector space)**

cites explicitly:

Definition 1505 (polynomial space $\mathbb{P}_k^d$).
**The proof of Lemma 1508** ($\mathbb{P}_k^d$ is nondecreasing sequence in $k$)

cites explicitly:

Lemma 1496 ($C_k^d$ are layers of $A_k^d$),

Definition 1505 (polynomial space $\mathbb{P}_k^d$).

**The proof of Lemma 1509** (constant and affine spaces $\mathbb{P}_0^d$ and $\mathbb{P}_1^d$)

cites explicitly:

Definition 1365 (canonic families),

Lemma 1399 (equivalent definition of affine map (finite dimension)),

Lemma 1497 (first multi-indices $A_k^d$),

Definition 1505 (polynomial space $\mathbb{P}_k^d$).

**The proof of Lemma 1511** (values of degree of polynomial)

cites explicitly:

Definition 1510 (degree of polynomial).

**The proof of Lemma 1512** (monomials of $C_k^d$ have degree $k$)

cites explicitly:

Definition 1486 (sets of multi-indices $A_k^d$ and $C_k^d$),

Definition 1503 (monomial in $d$ variables),

Definition 1510 (degree of polynomial).

**The proof of Lemma 1513** ($\mathbb{P}_k^d$ is space of degree at most $k$)

cites explicitly:

Lemma 1496 ($C_k^d$ are layers of $A_k^d$),

Definition 1505 (polynomial space $\mathbb{P}_k^d$),

Definition 1510 (degree of polynomial).

**The proof of Lemma 1514** (product of monomials)

cites explicitly:

Lemma 1481 (length of multi-indices is additive),

Definition 1503 (monomial in $d$ variables),

Lemma 1512 (monomials of $C_k^d$ have degree $k$).

**The proof of Lemma 1515** (product of monomial and polynomial)

cites explicitly:

Definition 1486 (sets of multi-indices $A_k^d$ and $C_k^d$),

Definition 1505 (polynomial space $\mathbb{P}_k^d$),

Lemma 1514 (product of monomials).

**The proof of Lemma 1516** (product of two polynomials)

cites explicitly:

Definition 1505 (polynomial space $\mathbb{P}_k^d$),

Lemma 1507 ($\mathbb{P}_k^d$ is vector space),

Lemma 1515 (product of monomial and polynomial).

**The proof of Lemma 1517** (partial derivative of monomials)

cites explicitly:

Definition 1503 (monomial in $d$ variables),

Lemma 1512 (monomials of $C_k^d$ have degree $k$).

**The proof of Lemma 1519** (partial derivative is linear)

cites explicitly:

Definition 1505 (polynomial space $\mathbb{P}_k^d$),

Lemma 1517 (partial derivative of monomials).
**The proof of Lemma 1520 (partial derivative of 0)**

cites explicitly:

Lemma 1519 (partial derivative is linear).

**The proof of Lemma 1521 (derivating more than degree is 0)**

cites explicitly:

Definition 1486 (sets of multi-indices $A_k^d$ and $C_k^d$),

Lemma 1517 (partial derivative of monomials).

**The proof of Lemma 1522 (partial derivative of monomials at 0)**

cites explicitly:

Definition 1482 (factorial of multi-indices),

Definition 1484 (Kronecker delta of multi-indices),

Lemma 1517 (partial derivative of monomials).

**The proof of Lemma 1523 (monomials are free in $\mathbb{P}_k^d$)**

cites explicitly:

Lemma 1483 (factorial of multi-index is positive),

Definition 1505 (polynomial space $\mathbb{P}_k^d$),

Lemma 1519 (partial derivative is linear),

Lemma 1520 (partial derivative of 0),

Lemma 1522 (partial derivative of monomials at 0).

**The proof of Lemma 1524 (monomials are a basis of $\mathbb{P}_k^d$)**

cites explicitly:

Definition 1505 (polynomial space $\mathbb{P}_k^d$),

Lemma 1523 (monomials are free in $\mathbb{P}_k^d$).

**The proof of Lemma 1525 (dimension of $\mathbb{P}_k^d$)**

cites explicitly:

Lemma 1498 (cardinal of $A_k^d$),

Lemma 1524 (monomials are a basis of $\mathbb{P}_k^d$).

**The proof of Lemma 1526 (isomorphism between $\mathbb{P}_0^d$ and $\mathbb{P}_0^{d-1}$)**

cites explicitly:

Statement(s) from [14],

Lemma 1374 (injectivity or surjectivity and dimension implies bijectivity),

Lemma 1509 (constant and affine spaces $\mathbb{P}_0^d$ and $\mathbb{P}_1^d$),

Lemma 1525 (dimension of $\mathbb{P}_k^d$).

**The proof of Lemma 1529 (decomposition of $\mathbb{P}_k^d$)**

cites explicitly:

Definition 1491 (slices $\tilde{S}_{k,i}^d$ and $\tilde{S}_{k,i}^d$),

Lemma 1493 (slices of $C_k^d$),

Lemma 1494 (cardinal of slices of $C_k^d$),

Lemma 1496 ($C_k^d$ are layers of $A_k^d$),

Definition 1505 (polynomial space $\mathbb{P}_k^d$),

Lemma 1507 ($\mathbb{P}_k^d$ is vector space),

Lemma 1508 ($\mathbb{P}_k^d$ is nondecreasing sequence in $k$),

Lemma 1509 (constant and affine spaces $\mathbb{P}_0^d$ and $\mathbb{P}_1^d$),

Lemma 1512 (monomials of $C_k^d$ have degree $k$),

Lemma 1523 (monomials are free in $\mathbb{P}_k^d$).

**The proof of Lemma 1531 (isomorphism between $\mathbb{P}_k^d$ and $\mathbb{P}_k^{d-1} \times \mathbb{P}_{k-1}^d$)**

cites explicitly:

Statement(s) from [14],
Lemma 1364 (*properties of the binomial coefficient*),

Lemma 1374 (*injectivity or surjectivity and dimension implies bijectivity*),

Lemma 1507 (*$\mathbb{P}_k^d$ is vector space*),

Lemma 1525 (*dimension of $\mathbb{P}_k^d$*),

Lemma 1529 (*decomposition of $\mathbb{P}_k^d$*).

**The proof of Lemma 1533 ($\mathbb{P}_k^d$ is nondecreasing sequence in $d$)**

cites explicitly:

Definition 1505 (*polynomial space $\mathbb{P}_k^d$*),

Lemma 1507 (*$\mathbb{P}_k^d$ is vector space*),

Lemma 1509 (*constant and affine spaces $\mathbb{P}_0^d$ and $\mathbb{P}_1^d$*),

Lemma 1531 (*isomorphism between $\mathbb{P}_k^d$ and $\mathbb{P}_k^{d-1} \times \mathbb{P}_{k-1}^d$*).

**The proof of Lemma 1534 (expression of $\mathbb{P}_k^d$ as polynomial of $x_d$)**

cites explicitly:

Lemma 1419 (*monomials are free in $\mathbb{P}_k^1$*),

Lemma 1526 (*isomorphism between $\mathbb{P}_0^d$ and $\mathbb{P}_0^{d-1}$*),

Lemma 1529 (*decomposition of $\mathbb{P}_k^d$*).

**The proof of Lemma 1537 (product of two polynomials (alternate proof))**

cites explicitly:

Statement(s) from [14],

Lemma 1360 (*strong double induction*),

Lemma 1422 (*product of two univariate polynomials*),

Lemma 1507 (*$\mathbb{P}_k^d$ is vector space*),

Lemma 1508 (*$\mathbb{P}_k^d$ is nondecreasing sequence in $k$*),

Lemma 1509 (*constant and affine spaces $\mathbb{P}_0^d$ and $\mathbb{P}_1^d$*),

Lemma 1515 (*product of monomial and polynomial*),

Lemma 1529 (*decomposition of $\mathbb{P}_k^d$*),

Lemma 1533 (*$\mathbb{P}_k^d$ is nondecreasing sequence in $d$*).

**The proof of Lemma 1538 (product of polynomials)**

cites explicitly:

Lemma 1516 (*product of two polynomials*),

Lemma 1537 (*product of two polynomials (alternate proof)*).

**The proof of Lemma 1539 (affine mapping of monomials is $\mathbb{P}_k^l$)**

cites explicitly:

Lemma 1399 (*equivalent definition of affine map (finite dimension)*),

Definition 1480 (*length of multi-indices*),

Definition 1486 (*sets of multi-indices $A_k^d$ and $C_k^d$*),

Definition 1503 (*monomial in $d$ variables*),

Lemma 1509 (*constant and affine spaces $\mathbb{P}_0^d$ and $\mathbb{P}_1^d$*),

Lemma 1538 (*product of polynomials*).

**The proof of Lemma 1540 (affine mapping of $\mathbb{P}_k^d$ is $\mathbb{P}_k^l$)**

cites explicitly:

Lemma 1496 (*$C_k^d$ are layers of $A_k^d$*),

Definition 1505 (*polynomial space $\mathbb{P}_k^d$*),

Lemma 1507 (*$\mathbb{P}_k^d$ is vector space*),

Lemma 1508 (*$\mathbb{P}_k^d$ is nondecreasing sequence in $k$*),

Lemma 1539 (*affine mapping of monomials is $\mathbb{P}_k^l$*).

**The proof of Lemma 1542 (reference Lagrange polynomials of $\mathbb{P}_1^d$ for $d=1$ are reference Lagrange polynomials of $\mathbb{P}_k^1$ for $k=1$)**

cites explicitly:
Definition 1448 (Lagrange polynomials of $\mathbb{P}_k^1$),

Definition 1453 (reference Lagrange nodes of $\mathbb{P}_k^1$),

Lemma 1455 (reference Lagrange basis of $\mathbb{P}_k^1$),

Definition 1541 (reference Lagrange polynomials of $\mathbb{P}_1^d$).

**The proof of Lemma 1543 (reference Lagrange polynomials is basis of $\mathbb{P}_1^d$)**

cites explicitly:

Lemma 1376 (free family of dim elements is basis),

Definition 1434 (family of reference points),

Lemma 1509 (constant and affine spaces $\mathbb{P}_0^d$ and $\mathbb{P}_1^d$),

Definition 1510 (degree of polynomial),

Lemma 1525 (dimension of $\mathbb{P}_k^d$),

Definition 1541 (reference Lagrange polynomials of $\mathbb{P}_1^d$).

**The proof of Lemma 1545 (differential of reference Lagrange polynomials)**

cites explicitly:

Lemma 1509 (constant and affine spaces $\mathbb{P}_0^d$ and $\mathbb{P}_1^d$),

Lemma 1543 (reference Lagrange polynomials is basis of $\mathbb{P}_1^d$).

**The proof of Lemma 1548 (geometric mapping for $d = 1$ is geometric mapping in dimension 1)**

cites explicitly:

Definition 1461 (geometric mapping in dimension 1),

Definition 1541 (reference Lagrange polynomials of $\mathbb{P}_1^d$),

Definition 1547 (geometric mapping).

**The proof of Lemma 1549 (reference geometric mapping is identity)**

cites explicitly:

Definition 1434 (family of reference points),

Definition 1541 (reference Lagrange polynomials of $\mathbb{P}_1^d$),

Definition 1547 (geometric mapping).

**The proof of Lemma 1550 (properties of geometric mapping)**

cites explicitly:

Statement(s) from [14],

Lemma 1374 (injectivity or surjectivity and dimension implies bijectivity),

Definition 1388 (affine map),

Lemma 1406 (injective affine map is zero linear kernel),

Lemma 1410 (inverse of affine map is affine map),

Definition 1411 (affinely independent family),

Definition 1437 (reference simplex),

Definition 1440 (simplex),

Definition 1541 (reference Lagrange polynomials of $\mathbb{P}_1^d$),

Lemma 1543 (reference Lagrange polynomials is basis of $\mathbb{P}_1^d$),

Definition 1547 (geometric mapping).

**The proof of Lemma 1551 (differential of geometric mapping)**

cites explicitly:

Lemma 1550 (properties of geometric mapping).

**The proof of Lemma 1552 (Lagrange polynomials of $\mathbb{P}_1^d$)**

cites explicitly:

Lemma 1550 (properties of geometric mapping).

**The proof of Lemma 1553 (Lagrange polynomials of reference vertices are reference Lagrange polynomials of $\mathbb{P}_1^d$)**

cites explicitly:
Lemma 1549 (reference geometric mapping is identity),

Lemma 1552 (Lagrange polynomials of $\mathbb{P}_1^d$).

**The proof of Lemma 1554 (Lagrange polynomials is basis of $\mathbb{P}_1^d$)**

cites explicitly:

Lemma 1376 (free family of dim elements is basis),

Lemma 1403 (affine maps are closed by composition),

Lemma 1509 (constant and affine spaces $\mathbb{P}_0^d$ and $\mathbb{P}_1^d$),

Lemma 1525 (dimension of $\mathbb{P}_k^d$),

Lemma 1543 (reference Lagrange polynomials is basis of $\mathbb{P}_1^d$),

Lemma 1550 (properties of geometric mapping),

Lemma 1552 (Lagrange polynomials of $\mathbb{P}_1^d$).

**The proof of Lemma 1555 (decomposition of $\mathbb{P}_1^d$ polynomial in Lagrange basis)**

cites explicitly:

Lemma 1554 (Lagrange polynomials is basis of $\mathbb{P}_1^d$).

**The proof of Lemma 1556 (differential of Lagrange polynomials $\mathbb{P}_1^d$)**

cites explicitly:

Lemma 1509 (constant and affine spaces $\mathbb{P}_0^d$ and $\mathbb{P}_1^d$),

Lemma 1545 (differential of reference Lagrange polynomials),

Lemma 1551 (differential of geometric mapping),

Lemma 1554 (Lagrange polynomials is basis of $\mathbb{P}_1^d$).

**The proof of Lemma 1557 (nontrivial simplex)**

cites explicitly:

Lemma 1439 (nontrivial reference simplex),

Lemma 1550 (properties of geometric mapping),

Lemma 1551 (differential of geometric mapping).

**The proof of Lemma 1559 (barycentric coordinate)**

cites explicitly:

Statement(s) from [14],

Lemma 1376 (free family of dim elements is basis),

Definition 1411 (affinely independent family),

Lemma 1412 (equivalent definition of affinely independent family).

**The proof of Lemma 1560 (Lagrange polynomials of $\mathbb{P}_1^d$ are barycentric coordinate)**

cites explicitly:

Definition 1547 (geometric mapping),

Lemma 1550 (properties of geometric mapping),

Lemma 1552 (Lagrange polynomials of $\mathbb{P}_1^d$),

Lemma 1559 (barycentric coordinate).

**The proof of Lemma 1561 (decomposition of $\mathbb{P}_1^d$ polynomial with barycentric coordinates)**

cites explicitly:

Lemma 1555 (decomposition of $\mathbb{P}_1^d$ polynomial in Lagrange basis),

Lemma 1560 (Lagrange polynomials of $\mathbb{P}_1^d$ are barycentric coordinate).

**The proof of Lemma 1563 (equivalent definition of face hyperplane)**

cites explicitly:

Statement(s) from [14],

Lemma 1387 (barycenter closure is affine subspace),

Lemma 1559 (barycentric coordinate),

Definition 1562 (face hyperplane).
**The proof of Lemma 1564 (reference face hyperplane)**

cites explicitly:

Statement(s) from [14],

Lemma 1436 (family of reference points is affinely independent),

Definition 1541 (reference Lagrange polynomials of $\mathbb{P}_1^d$),

Lemma 1553 (Lagrange polynomials of reference vertices are reference Lagrange polynomials of $\mathbb{P}_1^d$),

Lemma 1560 (Lagrange polynomials of $\mathbb{P}_1^d$ are barycentric coordinate),

Lemma 1563 (equivalent definition of face hyperplane).

**The proof of Lemma 1565 (face hyperplane is image of reference face hyperplane)**

cites explicitly:

Definition 1434 (family of reference points),

Definition 1541 (reference Lagrange polynomials of $\mathbb{P}_1^d$),

Definition 1547 (geometric mapping),

Definition 1562 (face hyperplane).

**The proof of Lemma 1567 (hyperface is included in face hyperplane)**

cites explicitly:

Lemma 1563 (equivalent definition of face hyperplane),

Definition 1566 (hyperface).

**The proof of Lemma 1569 (equivalent definition of $l$-face affine space)**

cites explicitly:

Lemma 1387 (barycenter closure is affine subspace),

Definition 1568 (l-face affine space).

**The proof of Lemma 1570 (d-face affine space is full space)**

cites explicitly:

Lemma 1376 (free family of dim elements is basis),

Definition 1411 (affinely independent family),

Lemma 1414 (affinely independent family is closed by sub-family),

Lemma 1569 (equivalent definition of l-face affine space).

**The proof of Lemma 1571 (0-face affine space is vertex)**

cites explicitly:

Lemma 1569 (equivalent definition of l-face affine space).

**The proof of Lemma 1573 (l-face is included in l-face affine space)**

cites explicitly:

Definition 1568 (l-face affine space),

Definition 1572 (l-face).

**The proof of Lemma 1574 (d-face is simplex)**

cites explicitly:

Definition 1440 (simplex),

Definition 1572 (l-face).

**The proof of Lemma 1576 ($(d-1)$-face is hyperface)**

cites explicitly:

Lemma 1368 (jump enumeration),

Lemma 1563 (equivalent definition of face hyperplane),

Definition 1566 (hyperface),

Definition 1568 (l-face affine space),

Definition 1572 (l-face).
**The proof of Lemma 1579 (geometric $d$-face mapping is geometric mapping)**

cites explicitly:

Definition 1547 (geometric mapping),

Definition 1578 (geometric $l$-face mapping).

**The proof of Lemma 1581 (properties of geometric $l$-face mapping)**

cites explicitly:

Statement(s) from [14],

Definition 1388 (affine map),

Lemma 1406 (injective affine map is zero linear kernel),

Lemma 1409 (inverse of affine submap is affine submap),

Lemma 1414 (affinely independent family is closed by sub-family),

Definition 1437 (reference simplex),

Definition 1541 (reference Lagrange polynomials of $\mathbb{P}_1^d$),

Lemma 1543 (reference Lagrange polynomials is basis of $\mathbb{P}_1^d$),

Definition 1568 (l-face affine space),

Lemma 1569 (equivalent definition of l-face affine space),

Definition 1572 (l-face),

Definition 1578 (geometric l-face mapping).

**The proof of Lemma 1582 (geometric $l$-face mapping of $\mathbb{P}_k^d$ is $\mathbb{P}_k^l$)**

cites explicitly:

Lemma 1540 (affine mapping of $\mathbb{P}_k^d$ is $\mathbb{P}_k^l$),

Lemma 1581 (properties of geometric $l$-face mapping).

**The proof of Lemma 1583 (geometric mapping of $\mathbb{P}_k^d$ is $\mathbb{P}_k^d$)**

cites explicitly:

Lemma 1540 (affine mapping of $\mathbb{P}_k^d$ is $\mathbb{P}_k^l$),

Lemma 1550 (properties of geometric mapping),

Lemma 1579 (geometric $d$-face mapping is geometric mapping),

Lemma 1582 (geometric $l$-face mapping of $\mathbb{P}_k^d$ is $\mathbb{P}_k^l$).

**The proof of Lemma 1584 (geometric hyperface mapping)**

cites explicitly:

Lemma 1576 ((d-1)-face is hyperface),

Lemma 1581 (properties of geometric $l$-face mapping).

**The proof of Lemma 1585 (hyperface geometric mapping of $\mathbb{P}_k^d$ is $\mathbb{P}_k^{d-1}$)**

cites explicitly:

Lemma 1576 ((d-1)-face is hyperface),

Lemma 1582 (geometric $l$-face mapping of $\mathbb{P}_k^d$ is $\mathbb{P}_k^l$).

**The proof of Lemma 1586 (geometric mapping with permutation)**

cites explicitly:

Lemma 1370 (image of ker is ker),

Lemma 1543 (reference Lagrange polynomials is basis of $\mathbb{P}_1^d$),

Lemma 1559 (barycentric coordinate),

Lemma 1560 (Lagrange polynomials of $\mathbb{P}_1^d$ are barycentric coordinate),

Lemma 1563 (equivalent definition of face hyperplane),

Lemma 1564 (reference face hyperplane),

Lemma 1570 (d-face affine space is full space),

Lemma 1574 (d-face is simplex),

Definition 1578 (geometric $l$-face mapping),

Lemma 1581 (properties of geometric $l$-face mapping).

**The proof of Lemma 1589 (Lagrange nodes of $\mathbb{P}_k^d$ for $d=1$ are Lagrange nodes of $\mathbb{P}_k^1$)**
cites explicitly:

Definition 1465 (Lagrange nodes of $\mathbb{P}_k^1$),

Lemma 1497 (first multi-indices $A_k^d$),

Definition 1588 (Lagrange nodes of $\mathbb{P}_k^d$).

**The proof of Lemma 1590 (number of Lagrange nodes of $\mathbb{P}_k^d$)**

cites explicitly:

Lemma 1498 (cardinal of $A_k^d$),

Lemma 1559 (barycentric coordinate),

Definition 1588 (Lagrange nodes of $\mathbb{P}_k^d$).

**The proof of Lemma 1591 (barycentric coordinates of Lagrange nodes of $\mathbb{P}_k^d$)**

cites explicitly:

Definition 1401 (isobarycenter),

Definition 1480 (length of multi-indices),

Definition 1486 (sets of multi-indices $A_k^d$ and $C_k^d$),

Lemma 1488 (indices are smaller than maximal length),

Lemma 1559 (barycentric coordinate),

Definition 1588 (Lagrange nodes of $\mathbb{P}_k^d$).

**The proof of Lemma 1592 (vertices are Lagrange nodes of $\mathbb{P}_k^d$)**

cites explicitly:

Definition 1365 (canonic families),

Definition 1480 (length of multi-indices),

Definition 1486 (sets of multi-indices $A_k^d$ and $C_k^d$),

Definition 1588 (Lagrange nodes of $\mathbb{P}_k^d$).

**The proof of Lemma 1593 (Lagrange nodes of $\mathbb{P}_1^d$ are vertices)**

cites explicitly:

Lemma 1497 (first multi-indices $A_k^d$),

Lemma 1592 (vertices are Lagrange nodes of $\mathbb{P}_k^d$).

**The proof of Lemma 1595 (equivalent definition of sub-vertices of Lagrange nodes of $\mathbb{P}_k^d$)**

cites explicitly:

Definition 1365 (canonic families),

Definition 1480 (length of multi-indices),

Definition 1486 (sets of multi-indices $A_k^d$ and $C_k^d$),

Lemma 1591 (barycentric coordinates of Lagrange nodes of $\mathbb{P}_k^d$),

Definition 1594 (sub-vertices of Lagrange nodes of $\mathbb{P}_k^d$).

**The proof of Lemma 1597 (sub-vertices are affinely independent)**

cites explicitly:

Definition 1411 (affinely independent family),

Lemma 1595 (equivalent definition of sub-vertices of Lagrange nodes of $\mathbb{P}_k^d$).

**The proof of Lemma 1598 ($\mathbb{P}_{k-1}^d$ sub-nodes of sub-vertices are some nodes of $\mathbb{P}_{k-1}^d$)**

cites explicitly:

Definition 1480 (length of multi-indices),

Lemma 1496 ($C_k^d$ are layers of $A_k^d$),

Lemma 1591 (barycentric coordinates of Lagrange nodes of $\mathbb{P}_k^d$),

Lemma 1595 (equivalent definition of sub-vertices of Lagrange nodes of $\mathbb{P}_k^d$),

Lemma 1597 (sub-vertices are affinely independent).

**The proof of Lemma 1599 (reference Lagrange nodes of $\mathbb{P}_k^d$)**

cites explicitly:

Definition 1434 (family of reference points),
Lemma 1435 (reference isobarycenter),

Definition 1588 (Lagrange nodes of $\mathbb{P}_k^d$),

Lemma 1591 (barycentric coordinates of Lagrange nodes of $\mathbb{P}_k^d$).

**The proof of Lemma 1600 (reference Lagrange nodes of $\mathbb{P}_k^d$ for $d=1$ are reference Lagrange nodes of $\mathbb{P}_k^1$)**

cites explicitly:

Definition 1365 (canonic families),

Definition 1453 (reference Lagrange nodes of $\mathbb{P}_k^1$),

Lemma 1497 (first multi-indices $A_k^d$),

Lemma 1599 (reference Lagrange nodes of $\mathbb{P}_k^d$).

**The proof of Lemma 1601 (equivalent definition of reference Lagrange nodes of $\mathbb{P}_k^d$)**

cites explicitly:

Definition 1365 (canonic families),

Lemma 1435 (reference isobarycenter),

Lemma 1599 (reference Lagrange nodes of $\mathbb{P}_k^d$).

**The proof of Lemma 1602 (number of reference Lagrange nodes of $\mathbb{P}_k^d$)**

cites explicitly:

Lemma 1436 (family of reference points is affinely independent),

Lemma 1498 (cardinal of $A_k^d$),

Lemma 1559 (barycentric coordinate),

Lemma 1599 (reference Lagrange nodes of $\mathbb{P}_k^d$).

**The proof of Lemma 1604 (Lagrange nodes of $\mathbb{P}_k^d$ are image of reference)**

cites explicitly:

Lemma 1550 (properties of geometric mapping),

Definition 1588 (Lagrange nodes of $\mathbb{P}_k^d$),

Lemma 1601 (equivalent definition of reference Lagrange nodes of $\mathbb{P}_k^d$).

**The proof of Lemma 1605 (face hyperplanes of Lagrange nodes of $\mathbb{P}_k^d$)**

cites explicitly:

Definition 1486 (sets of multi-indices $A_k^d$ and $C_k^d$),

Lemma 1495 (cardinal of $C_k^d$),

Lemma 1498 (cardinal of $A_k^d$),

Lemma 1500 (cardinal of $C_k^d$ and $A_k^{d-1}$),

Lemma 1501 (cardinal of $A_{k,i}^d$ and $A_k^{d-1}$),

Lemma 1563 (equivalent definition of face hyperplane),

Lemma 1590 (number of Lagrange nodes of $\mathbb{P}_k^d$),

Lemma 1591 (barycentric coordinates of Lagrange nodes of $\mathbb{P}_k^d$).

**The proof of Lemma 1607 (image of nodes by geometric hyperface mapping)**

cites explicitly:

Lemma 1400 (affine map preserves barycenter),

Definition 1486 (sets of multi-indices $A_k^d$ and $C_k^d$),

Lemma 1500 (cardinal of $C_k^d$ and $A_k^{d-1}$),

Lemma 1501 (cardinal of $A_{k,i}^d$ and $A_k^{d-1}$),

Lemma 1584 (geometric hyperface mapping),

Lemma 1591 (barycentric coordinates of Lagrange nodes of $\mathbb{P}_k^d$),

Lemma 1599 (reference Lagrange nodes of $\mathbb{P}_k^d$).

**The proof of Lemma 1609 (Lagrange linear forms for $\mathbb{P}_k^d$ for $d=1$ are Lagrange linear forms for $\mathbb{P}_k^1$)**

cites explicitly:

Definition 1472 (Lagrange linear forms for $\mathbb{P}_k^1$),
Lemma 1497 (first multi-indices $A_k^d$),

Lemma 1589 (Lagrange nodes of $\mathbb{P}_k^d$ for $d = 1$ are Lagrange nodes of $\mathbb{P}_k^1$),

Definition 1608 (Lagrange linear forms for $\mathbb{P}_k^d$).

**The proof of Lemma 1610 (Lagrange linear forms of $\mathbb{P}_k^d$ are linear)**

cites explicitly:

Definition 1608 (Lagrange linear forms for $\mathbb{P}_k^d$).

**The proof of Lemma 1611 (cardinal of Lagrange linear forms of $\mathbb{P}_k^d$)**

cites explicitly:

Lemma 1590 (number of Lagrange nodes of $\mathbb{P}_k^d$),

Definition 1608 (Lagrange linear forms for $\mathbb{P}_k^d$).

**The proof of Lemma 1613 (reference Lagrange linear forms for $\mathbb{P}_k^d$ for $d = 1$ are reference Lagrange linear forms for $\mathbb{P}_k^1$)**

cites explicitly:

Definition 1456 (reference Lagrange linear forms for $\mathbb{P}_k^1$),

Lemma 1497 (first multi-indices $A_k^d$),

Lemma 1600 (reference Lagrange nodes of $\mathbb{P}_k^d$ for $d = 1$ are reference Lagrange nodes of $\mathbb{P}_k^1$),

Definition 1612 (reference Lagrange linear forms for $\mathbb{P}_k^d$).

**The proof of Lemma 1614 (Lagrange linear forms of $\mathbb{P}_k^d$ are images of reference)**

cites explicitly:

Lemma 1604 (Lagrange nodes of $\mathbb{P}_k^d$ are image of reference),

Definition 1608 (Lagrange linear forms for $\mathbb{P}_k^d$),

Definition 1612 (reference Lagrange linear forms for $\mathbb{P}_k^d$).

**The proof of Lemma 1615 (Lagrange linear forms for $\mathbb{P}_0^d$ are injective)**

cites explicitly:

Statement(s) from [14],

Definition 1424 (finite element triple),

Lemma 1509 (constant and affine spaces $\mathbb{P}_0^d$ and $\mathbb{P}_1^d$),

Definition 1608 (Lagrange linear forms for $\mathbb{P}_k^d$).

**The proof of Lemma 1617 (unisolvence of $\mathbb{P}_0^d$)**

cites explicitly:

Lemma 1364 (properties of the binomial coefficient),

Lemma 1426 (injectivity implies unisolvence),

Lemma 1525 (dimension of $\mathbb{P}_k^d$),

Lemma 1611 (cardinal of Lagrange linear forms of $\mathbb{P}_k^d$),

Lemma 1615 (Lagrange linear forms for $\mathbb{P}_0^d$ are injective).

**The proof of Lemma 1618 (decomposition of $\mathbb{P}_1^d$ polynomial with $\sigma_\alpha$)**

cites explicitly:

Lemma 1555 (decomposition of $\mathbb{P}_1^d$ polynomial in Lagrange basis),

Lemma 1561 (decomposition of $\mathbb{P}_1^d$ polynomial with barycentric coordinates),

Lemma 1593 (Lagrange nodes of $\mathbb{P}_1^d$ are vertices),

Definition 1608 (Lagrange linear forms for $\mathbb{P}_k^d$).

**The proof of Lemma 1619 (Lagrange linear forms for $\mathbb{P}_1^d$ are injective)**

cites explicitly:

Statement(s) from [14],

Definition 1424 (finite element triple),

Lemma 1497 (first multi-indices $A_k^d$),

Lemma 1618 (decomposition of $\mathbb{P}_1^d$ polynomial with $\sigma_\alpha$).
**The proof of Lemma 1620 (unisolvence of $\mathbb{P}_1^d$)**

cites explicitly:

Lemma 1364 (*properties of the binomial coefficient*),

Lemma 1426 (*injectivity implies unisolvence*),

Lemma 1525 (*dimension of $\mathbb{P}_k^d$*),

Lemma 1611 (*cardinal of Lagrange linear forms of $\mathbb{P}_k^d$*),

Lemma 1619 (*Lagrange linear forms for $\mathbb{P}_1^d$ are injective*).

**The proof of Lemma 1621 (factorization of zero polynomial on last reference hyper-plane)**

cites explicitly:

Statement(s) from [14],

Lemma 1529 (*decomposition of $\mathbb{P}_k^d$*),

Definition 1541 (*reference Lagrange polynomials of $\mathbb{P}_1^d$*),

Lemma 1564 (*reference face hyperplane*).

**The proof of Lemma 1623 (factorization of zero polynomial on hyperplane $\mathbb{P}_k^d$)**

cites explicitly:

Statement(s) from [14],

Lemma 1366 (*circular permutation*),

Lemma 1410 (*inverse of affine map is affine map*),

Lemma 1540 (*affine mapping of $\mathbb{P}_k^d$ is $\mathbb{P}_k^l$*),

Lemma 1563 (*equivalent definition of face hyperplane*),

Lemma 1586 (*geometric mapping with permutation*),

Lemma 1621 (*factorization of zero polynomial on last reference hyperplane*).

**The proof of Lemma 1625 (Lagrange linear forms for $\mathbb{P}_k^d$ are injective)**

cites explicitly:

Statement(s) from [14],

Lemma 1358 (*double induction by diagonal*),

Lemma 1413 (*affinely independent family of 2 elements*),

Definition 1424 (*finite element triple*),

Lemma 1436 (*family of reference points is affinely independent*),

Lemma 1476 (*Lagrange linear forms for $\mathbb{P}_k^1$ are injective*),

Lemma 1496 (*$C_k^d$ are layers of $\mathcal{A}_k^d$*),

Lemma 1500 (*cardinal of $C_k^d$ and $\mathcal{A}_k^{d-1}$*),

Lemma 1563 (*equivalent definition of face hyperplane*),

Lemma 1584 (*geometric hyperface mapping*),

Lemma 1585 (*hyperface geometric mapping of $\mathbb{P}_k^d$ is $\mathbb{P}_k^{d-1}$*),

Definition 1588 (*Lagrange nodes of $\mathbb{P}_k^d$*),

Lemma 1597 (*sub-vertices are affinely independent*),

Lemma 1598 (*$\mathbb{P}_{k-1}^d$ sub-nodes of sub-vertices are some nodes of $\mathbb{P}_{k-1}^d$*),

Lemma 1605 (*face hyperplanes of Lagrange nodes of $\mathbb{P}_k^d$*),

Lemma 1607 (*image of nodes by geometric hyperface mapping*),

Definition 1608 (*Lagrange linear forms for $\mathbb{P}_k^d$*),

Lemma 1609 (*Lagrange linear forms for $\mathbb{P}_k^d$ for $d=1$ are Lagrange linear forms for $\mathbb{P}_k^1$*),

Lemma 1619 (*Lagrange linear forms for $\mathbb{P}_1^d$ are injective*),

Lemma 1623 (*factorization of zero polynomial on hyperplane $\mathbb{P}_k^d$*).

**The proof of Theorem 1626 (unisolvence of $\mathbb{P}_k^d$)**

cites explicitly:

Lemma 1426 (*injectivity implies unisolvence*),

Lemma 1525 (*dimension of $\mathbb{P}_k^d$*),

Definition 1608 (*Lagrange linear forms for $\mathbb{P}_k^d$*),
Lemma 1611 (cardinal of Lagrange linear forms of $\mathbb{P}_k^d$),

Lemma 1625 (Lagrange linear forms for $\mathbb{P}_k^d$ are injective).

**The proof of Lemma 1628 (face unisolvence of $\mathbb{P}_k^d$)**

cites explicitly:

Statement(s) from [14],

Definition 1424 (finite element triple),

Lemma 1436 (family of reference points is affinely independent),

Lemma 1500 (cardinal of $C_k^d$ and $A_k^{d-1}$),

Lemma 1501 (cardinal of $A_{k,i}^d$ and $A_k^{d-1}$),

Lemma 1584 (geometric hyperface mapping),

Lemma 1585 (hyperface geometric mapping of $\mathbb{P}_k^d$ is $\mathbb{P}_k^{d-1}$),

Lemma 1605 (face hyperplanes of Lagrange nodes of $\mathbb{P}_k^d$),

Lemma 1607 (image of nodes by geometric hyperface mapping),

Definition 1608 (Lagrange linear forms for $\mathbb{P}_k^d$),

Lemma 1625 (Lagrange linear forms for $\mathbb{P}_k^d$ are injective).

**The proof of Theorem 1629 ($\text{Log}\mathbb{P}_k^d$ Lagrange finite element)**

cites explicitly:

Definition 1424 (finite element triple),

Lemma 1525 (dimension of $\mathbb{P}_k^d$),

Lemma 1557 (nontrivial simplex),

Lemma 1617 (unisolvence of $\mathbb{P}_0^d$),

Theorem 1626 (unisolvence of $\mathbb{P}_k^d$).

**The proof of Lemma 1630 ($\text{Log}\mathbb{P}_k^d$ Lagrange finite element for $d=1$ is Lagrange finite element on a current segment)**

cites explicitly:

Theorem 1478 ($\text{Log}\mathbb{P}_k^1$ Lagrange finite element),

Lemma 1609 (Lagrange linear forms for $\mathbb{P}_k^d$ for $d=1$ are Lagrange linear forms for $\mathbb{P}_k^1$),

Theorem 1629 ($\text{Log}\mathbb{P}_k^d$ Lagrange finite element).

**The proof of Theorem 1631 ($\text{Log}\hat{\mathbb{P}}_k^d$ reference Lagrange finite element)**

cites explicitly:

Lemma 1436 (family of reference points is affinely independent),

Lemma 1443 (simplex of reference vertices is reference simplex),

Lemma 1599 (reference Lagrange nodes of $\mathbb{P}_k^d$),

Definition 1612 (reference Lagrange linear forms for $\mathbb{P}_k^d$),

Theorem 1629 ($\text{Log}\mathbb{P}_k^d$ Lagrange finite element).

**The proof of Lemma 1632 ($\text{Log}\hat{\mathbb{P}}_k^d$ reference Lagrange finite element for $d=1$ is Lagrange finite element on the reference segment)**

cites explicitly:

Theorem 1460 ($\text{Log}\hat{\mathbb{P}}_k^1$ reference Lagrange finite element),

Lemma 1613 (reference Lagrange linear forms for $\mathbb{P}_k^d$ for $d=1$ are reference Lagrange linear forms for $\mathbb{P}_k^1$),

Theorem 1631 ($\text{Log}\hat{\mathbb{P}}_k^d$ reference Lagrange finite element).
# Appendix C

## Is explicitly cited in the proof of...

This appendix gathers the explicit citations that appear in the proof of results (lemmas and theorems) for each statement listed in Appendix A. Statements from [14, 15] are anonymized.

The dual graph is described in Appendix B.

Printing is not advised!

**Statement(s) from [14]**

are explicitly cited in the proof of:

* Lemma 1369 (image of ker is included in ker),

* Lemma 1370 (image of ker is ker),

* Lemma 1372 (vector subspace is invariant by translation),

* Lemma 1373 (range of linear map is vector subspace),

* Lemma 1374 (injectivity or surjectivity and dimension implies bijectivity),

* Lemma 1375 (inverse of isomorphism is linear map),

* Lemma 1380 (origin is in affine subspace),

* Lemma 1381 (equivalent definition of affine subspace),

* Lemma 1382 (affine subspace is invariant by change of origin),

* Lemma 1385 (affine subspace plus vector subspace is affine subspace),

* Lemma 1386 (closed under barycenter is affine subspace),

* Lemma 1387 (barycenter closure is affine subspace),

* Lemma 1391 (change of origin in affine map),

* Lemma 1396 (space of affine maps),

* Lemma 1400 (affine map preserves barycenter),

* Lemma 1402 (affine map preserves isobarycenter),

* Lemma 1403 (affine maps are closed by composition),

* Lemma 1404 (continuous affine map is continuous linear map),

* Lemma 1405 (injective affine submap is zero linear kernel),

* Lemma 1407 (surjective affine submap is full linear range),

* Lemma 1409 (inverse of affine submap is affine submap),

* Lemma 1412 (equivalent definition of affinely independent family),

* Lemma 1458 (reference Lagrange linear forms for $\mathbb{P}_k^1$ are injective),

* Lemma 1476 (Lagrange linear forms for $\mathbb{P}_k^1$ are injective),

* Lemma 1526 (isomorphism between $\mathbb{P}_0^d$ and $\mathbb{P}_0^{d-1}$),

* Lemma 1531 (isomorphism between $\mathbb{P}_k^d$ and $\mathbb{P}_k^{d-1} \times \mathbb{P}_{k-1}^d$),

* Lemma 1537 (product of two polynomials (alternate proof)),

* Lemma 1550 (properties of geometric mapping),

* Lemma 1559 (barycentric coordinate),

* Lemma 1563 (equivalent definition of face hyperplane),

* Lemma 1564 (reference face hyperplane),
Lemma 1581 (*properties of geometric l-face mapping*),

Lemma 1615 (*Lagrange linear forms for $\mathbb{P}_0^d$ are injective*),

Lemma 1619 (*Lagrange linear forms for $\mathbb{P}_1^d$ are injective*),

Lemma 1621 (*factorization of zero polynomial on last reference hyperplane*),

Lemma 1623 (*factorization of zero polynomial on hyperplane $\mathbb{P}_k^d$*),

Lemma 1625 (*Lagrange linear forms for $\mathbb{P}_k^d$ are injective*),

Lemma 1628 (*face unisolvence of $\mathbb{P}_k^d$*).

**Statement(s) from [15]**

is not yet used.

**Lemma 1358 (double induction by diagonal)**

is explicitly cited in the proof of:

Lemma 1625 (*Lagrange linear forms for $\mathbb{P}_k^d$ are injective*).

**Lemma 1360 (strong double induction)**

is explicitly cited in the proof of:

Lemma 1537 (*product of two polynomials (alternate proof)*).

**Definition 1362 (binomial coefficient)**

is explicitly cited in the proof of:

Lemma 1364 (*properties of the binomial coefficient*),

Lemma 1495 (*cardinal of $C_k^d$*).

**Lemma 1364 (properties of the binomial coefficient)**

is explicitly cited in the proof of:

Lemma 1446 (*cardinal of $A_k^1$*),

Lemma 1495 (*cardinal of $C_k^d$*),

Lemma 1498 (*cardinal of $A_k^{d'}$*),

Lemma 1531 (*isomorphism between $\mathbb{P}_k^d$ and $\mathbb{P}_{k-1}^{d-1} \times \mathbb{P}_k^{d-1}$*),

Lemma 1617 (*unisolvence of $\mathbb{P}_0^d$*),

Lemma 1620 (*unisolvence of $\mathbb{P}_1^d$*).

**Definition 1365 (canonic families)**

is explicitly cited in the proof of:

Lemma 1436 (*family of reference points is affinely independent*),

Lemma 1489 (*first $C_k^d$*),

Lemma 1509 (*constant and affine spaces $\mathbb{P}_0^d$ and $\mathbb{P}_1^d$*),

Lemma 1592 (*vertices are Lagrange nodes of $\mathbb{P}_k^d$*),

Lemma 1595 (*equivalent definition of sub-vertices of Lagrange nodes of $\mathbb{P}_k^d$*),

Lemma 1600 (*reference Lagrange nodes of $\mathbb{P}_k^d$ for $d = 1$ are reference Lagrange nodes of $\mathbb{P}_k^1$*),

Lemma 1601 (*equivalent definition of reference Lagrange nodes of $\mathbb{P}_k^d$*).

**Lemma 1366 (circular permutation)**

is explicitly cited in the proof of:

Lemma 1623 (*factorization of zero polynomial on hyperplane $\mathbb{P}_k^d$*).

**Lemma 1367 (transposition)**

is not yet used.

**Lemma 1368 (jump enumeration)**

is explicitly cited in the proof of:

Lemma 1576 (($d-1$)-face is hyperface).

**Lemma 1369 (image of ker is included in ker)**

is explicitly cited in the proof of:

Lemma 1370 (*image of ker is ker*).
**Lemma 1370 (image of ker is ker)**

is explicitly cited in the proof of:

Lemma 1586 (*geometric mapping with permutation*).

**Lemma 1372 (vector subspace is invariant by translation)**

is explicitly cited in the proof of:

Lemma 1381 (*equivalent definition of affine subspace*),

Lemma 1382 (*affine subspace is invariant by change of origin*),

Lemma 1383 (*vector subspace is affine subspace*).

**Lemma 1373 (range of linear map is vector subspace)**

is explicitly cited in the proof of:

Lemma 1393 (*range of affine map is affine subspace*).

**Lemma 1374 (injectivity or surjectivity and dimension implies bijectivity)**

is explicitly cited in the proof of:

Lemma 1426 (*injectivity implies unisolvence*),

Lemma 1526 (*isomorphism between $\mathbb{P}_0^d$ and $\mathbb{P}_{0-1}^d$*),

Lemma 1531 (*isomorphism between $\mathbb{P}_k^d$ and $\mathbb{P}_{k-1}^{d-1}$ × $\mathbb{P}_{k-1}^d$*),

Lemma 1550 (*properties of geometric mapping*).

**Lemma 1375 (inverse of isomorphism is linear map)**

is explicitly cited in the proof of:

Lemma 1409 (*inverse of affine submap is affine submap*).

**Lemma 1376 (free family of dim elements is basis)**

is explicitly cited in the proof of:

Lemma 1420 (*dimension of $\mathbb{P}_k^1$*),

Lemma 1428 (*degrees of freedom are basis*),

Lemma 1449 (*Lagrange polynomials is basis of $\mathbb{P}_k^1$*),

Lemma 1543 (*reference Lagrange polynomials is basis of $\mathbb{P}_1^d$*),

Lemma 1554 (*Lagrange polynomials is basis of $\mathbb{P}_1^d$*),

Lemma 1559 (*barycentric coordinate*),

Lemma 1570 (*d-face affine space is full space*).

**Definition 1379 (affine subspace)**

is explicitly cited in the proof of:

Lemma 1380 (*origin is in affine subspace*),

Lemma 1381 (*equivalent definition of affine subspace*),

Lemma 1382 (*affine subspace is invariant by change of origin*),

Lemma 1385 (*affine subspace plus vector subspace is affine subspace*),

Lemma 1386 (*closed under barycenter is affine subspace*),

Lemma 1387 (*barycenter closure is affine subspace*),

Lemma 1390 (*equivalent definition of affine map*),

Lemma 1391 (*change of origin in affine map*),

Lemma 1393 (*range of affine map is affine subspace*),

Lemma 1398 (*affine submap*),

Lemma 1407 (*surjective affine submap is full linear range*),

Lemma 1409 (*inverse of affine submap is affine submap*).

**Lemma 1380 (origin is in affine subspace)**

is not yet used.

**Lemma 1381 (equivalent definition of affine subspace)**

is explicitly cited in the proof of:

Lemma 1382 (*affine subspace is invariant by change of origin*),

Lemma 1383 (*vector subspace is affine subspace*),
Lemma 1385 (affine subspace plus vector subspace is affine subspace),

Lemma 1386 (closed under barycenter is affine subspace),

Lemma 1390 (equivalent definition of affine map),

Lemma 1391 (change of origin in affine map),

Lemma 1407 (surjective affine submap is full linear range),

Lemma 1409 (inverse of affine submap is affine submap).

**Lemma 1382 (affine subspace is invariant by change of origin)**

is explicitly cited in the proof of:

Lemma 1397 (output restriction of affine map),

Lemma 1407 (surjective affine submap is full linear range).

**Lemma 1383 (vector subspace is affine subspace)**

is not yet used.

**Lemma 1385 (affine subspace plus vector subspace is affine subspace)**

is not yet used.

**Lemma 1386 (closed under barycenter is affine subspace)**

is not yet used.

**Lemma 1387 (barycenter closure is affine subspace)**

is explicitly cited in the proof of:

Lemma 1563 (equivalent definition of face hyperplane),

Lemma 1569 (equivalent definition of l-face affine space).

**Definition 1388 (affine map)**

is explicitly cited in the proof of:

Lemma 1390 (equivalent definition of affine map),

Lemma 1391 (change of origin in affine map),

Lemma 1393 (range of affine map is affine subspace),

Lemma 1398 (affine submap),

Lemma 1399 (equivalent definition of affine map (finite dimension)),

Lemma 1400 (affine map preserves barycenter),

Lemma 1403 (affine maps are closed by composition),

Lemma 1407 (surjective affine submap is full linear range),

Lemma 1409 (inverse of affine submap is affine submap),

Lemma 1550 (properties of geometric mapping),

Lemma 1581 (properties of geometric l-face mapping).

**Lemma 1390 (equivalent definition of affine map)**

is explicitly cited in the proof of:

Lemma 1403 (affine maps are closed by composition),

Lemma 1404 (continuous affine map is continuous linear map),

Lemma 1407 (surjective affine submap is full linear range),

Lemma 1409 (inverse of affine submap is affine submap).

**Lemma 1391 (change of origin in affine map)**

is explicitly cited in the proof of:

Lemma 1398 (affine submap).

**Lemma 1393 (range of affine map is affine subspace)**

is explicitly cited in the proof of:

Lemma 1397 (output restriction of affine map).

**Definition 1395 (set of affine maps)**

is explicitly cited in the proof of:
Lemma 1396 (space of affine maps),

Lemma 1397 (output restriction of affine map),

Lemma 1398 (affine submap),

Lemma 1403 (affine maps are closed by composition).

**Lemma 1396 (space of affine maps)**

is explicitly cited in the proof of:

Lemma 1418 ($\mathbb{P}_k^1$ is space of degree at most $k$).

**Lemma 1397 (output restriction of affine map)**

is explicitly cited in the proof of:

Lemma 1398 (affine submap).

**Lemma 1398 (affine submap)**

is explicitly cited in the proof of:

Lemma 1405 (injective affine submap is zero linear kernel).

**Lemma 1399 (equivalent definition of affine map (finite dimension))**

is explicitly cited in the proof of:

Lemma 1418 ($\mathbb{P}_k^1$ is space of degree at most $k$),

Lemma 1462 (properties of geometric mapping in dimension 1),

Lemma 1509 (constant and affine spaces $\mathbb{P}_0^d$ and $\mathbb{P}_1^d$),

Lemma 1539 (affine mapping of monomials is $\mathbb{P}_k^l$).

**Lemma 1400 (affine map preserves barycenter)**

is explicitly cited in the proof of:

Lemma 1402 (affine map preserves isobarycenter),

Lemma 1607 (image of nodes by geometric hyperface mapping).

**Definition 1401 (isobarycenter)**

is explicitly cited in the proof of:

Lemma 1402 (affine map preserves isobarycenter),

Lemma 1435 (reference isobarycenter),

Lemma 1591 (barycentric coordinates of Lagrange nodes of $\mathbb{P}_k^d$).

**Lemma 1402 (affine map preserves isobarycenter)**

is not yet used.

**Lemma 1403 (affine maps are closed by composition)**

is explicitly cited in the proof of:

Lemma 1554 (Lagrange polynomials is basis of $\mathbb{P}_1^d$).

**Lemma 1404 (continuous affine map is continuous linear map)**

is not yet used.

**Lemma 1405 (injective affine submap is zero linear kernel)**

is explicitly cited in the proof of:

Lemma 1406 (injective affine map is zero linear kernel),

Lemma 1409 (inverse of affine submap is affine submap).

**Lemma 1406 (injective affine map is zero linear kernel)**

is explicitly cited in the proof of:

Lemma 1550 (properties of geometric mapping),

Lemma 1581 (properties of geometric l-face mapping).

**Lemma 1407 (surjective affine submap is full linear range)**

is explicitly cited in the proof of:

Lemma 1408 (surjective affine map is full linear range),

Lemma 1409 (inverse of affine submap is affine submap).
**Lemma 1408 (surjective affine map is full linear range)**

is not yet used.

**Lemma 1409 (inverse of affine submap is affine submap)**

is explicitly cited in the proof of:

Lemma 1410 (inverse of affine map is affine map),

Lemma 1581 (properties of geometric $l$-face mapping).

**Lemma 1410 (inverse of affine map is affine map)**

is explicitly cited in the proof of:

Lemma 1462 (properties of geometric mapping in dimension 1),

Lemma 1550 (properties of geometric mapping),

Lemma 1623 (factorization of zero polynomial on hyperplane $\mathbb{P}_k^d$).

**Definition 1411 (affinely independent family)**

is explicitly cited in the proof of:

Lemma 1412 (equivalent definition of affinely independent family),

Lemma 1413 (affinely independent family of 2 elements),

Lemma 1414 (affinely independent family is closed by sub-family),

Lemma 1436 (family of reference points is affinely independent),

Lemma 1550 (properties of geometric mapping),

Lemma 1559 (barycentric coordinate),

Lemma 1570 (d-face affine space is full space),

Lemma 1597 (sub-vertices are affinely independent).

**Lemma 1412 (equivalent definition of affinely independent family)**

is explicitly cited in the proof of:

Lemma 1414 (affinely independent family is closed by sub-family),

Lemma 1559 (barycentric coordinate).

**Lemma 1413 (affinely independent family of 2 elements)**

is explicitly cited in the proof of:

Lemma 1625 (Lagrange linear forms for $\mathbb{P}_k^d$ are injective).

**Lemma 1414 (affinely independent family is closed by sub-family)**

is explicitly cited in the proof of:

Lemma 1570 (d-face affine space is full space),

Lemma 1581 (properties of geometric $l$-face mapping).

**Definition 1416 (monomial of a single variable)**

is explicitly cited in the proof of:

Lemma 1418 ($\mathbb{P}_k^1$ is space of degree at most $k$),

Lemma 1504 (monomial in $d$ variables for $d = 1$ is monomial of a single variable).

**Definition 1417 (polynomial space $\mathbb{P}_k^1$)**

is explicitly cited in the proof of:

Lemma 1418 ($\mathbb{P}_k^1$ is space of degree at most $k$),

Lemma 1420 (dimension of $\mathbb{P}_k^1$),

Lemma 1449 (Lagrange polynomials is basis of $\mathbb{P}_k^1$),

Lemma 1471 (geometric mapping of $\mathbb{P}_k^1$ is $\mathbb{P}_k^1$),

Lemma 1506 (polynomial space $\mathbb{P}_k^d$ for $d = 1$ is $\mathbb{P}_k^1$).

**Lemma 1418 ($\mathbb{P}_k^1$ is space of degree at most $k$)**

is explicitly cited in the proof of:

Lemma 1449 (Lagrange polynomials is basis of $\mathbb{P}_k^1$),

Lemma 1462 (properties of geometric mapping in dimension 1).
**Lemma 1419 (monomials are free in $\mathbb{P}_k^1$)**

is explicitly cited in the proof of:

Lemma 1420 (dimension of $\mathbb{P}_k^1$),

Lemma 1534 (expression of $\mathbb{P}_k^d$ as polynomial of $x_d$).

**Lemma 1420 (dimension of $\mathbb{P}_k^1$)**

is explicitly cited in the proof of:

Lemma 1449 (Lagrange polynomials is basis of $\mathbb{P}_k^1$),

Lemma 1459 (unisolvence of $\mathbb{P}_k^1$ (reference)),

Theorem 1460 ($^{\text{Log}}\hat{\mathbb{P}}_k^1$ reference Lagrange finite element),

Lemma 1477 (unisolvence of $\mathbb{P}_k^1$),

Theorem 1478 ($^{\text{Log}}\mathbb{P}_k^1$ Lagrange finite element).

**Lemma 1422 (product of two univariate polynomials)**

is explicitly cited in the proof of:

Lemma 1449 (Lagrange polynomials is basis of $\mathbb{P}_k^1$),

Lemma 1537 (product of two polynomials (alternate proof)).

**Definition 1424 (finite element triple)**

is explicitly cited in the proof of:

Lemma 1426 (injectivity implies unisolvence),

Lemma 1458 (reference Lagrange linear forms for $\mathbb{P}_k^1$ are injective),

Theorem 1460 ($^{\text{Log}}\hat{\mathbb{P}}_k^1$ reference Lagrange finite element),

Lemma 1476 (Lagrange linear forms for $\mathbb{P}_k^1$ are injective),

Theorem 1478 ($^{\text{Log}}\mathbb{P}_k^1$ Lagrange finite element),

Lemma 1615 (Lagrange linear forms for $\mathbb{P}_0^d$ are injective),

Lemma 1619 (Lagrange linear forms for $\mathbb{P}_1^d$ are injective),

Lemma 1625 (Lagrange linear forms for $\mathbb{P}_k^d$ are injective),

Lemma 1628 (face unisolvence of $\mathbb{P}_k^d$),

Theorem 1629 ($^{\text{Log}}\mathbb{P}_k^d$ Lagrange finite element).

**Lemma 1426 (injectivity implies unisolvence)**

is explicitly cited in the proof of:

Lemma 1459 (unisolvence of $\mathbb{P}_k^1$ (reference)),

Lemma 1477 (unisolvence of $\mathbb{P}_k^1$),

Lemma 1617 (unisolvence of $\mathbb{P}_0^d$),

Lemma 1620 (unisolvence of $\mathbb{P}_1^d$),

Theorem 1626 (unisolvence of $\mathbb{P}_k^d$).

**Lemma 1427 (dimension of approximation space)**

is explicitly cited in the proof of:

Lemma 1428 (degrees of freedom are basis).

**Lemma 1428 (degrees of freedom are basis)**

is not yet used.

**Definition 1430 (shape function)**

is not yet used.

**Definition 1432 (family of points)**

is not yet used.

**Definition 1434 (family of reference points)**

is explicitly cited in the proof of:

Lemma 1435 (reference isobarycenter),

Lemma 1436 (family of reference points is affinely independent),
Lemma 1443 (*simplex of reference vertices is reference simplex*),

Lemma 1543 (*reference Lagrange polynomials is basis of $\mathbb{P}_1^d$*),

Lemma 1549 (*reference geometric mapping is identity*),

Lemma 1565 (*face hyperplane is image of reference face hyperplane*),

Lemma 1599 (*reference Lagrange nodes of $\mathbb{P}_k^d$*).

**Lemma 1435 (reference isobarycenter)**

is explicitly cited in the proof of:

Lemma 1599 (*reference Lagrange nodes of $\mathbb{P}_k^d$*),

Lemma 1601 (*equivalent definition of reference Lagrange nodes of $\mathbb{P}_k^d$*).

**Lemma 1436 (family of reference points is affinely independent)**

is explicitly cited in the proof of:

Lemma 1564 (*reference face hyperplane*),

Lemma 1602 (*number of reference Lagrange nodes of $\mathbb{P}_k^d$*),

Lemma 1625 (*Lagrange linear forms for $\mathbb{P}_k^d$ are injective*),

Lemma 1628 (*face unisolvence of $\mathbb{P}_k^d$*),

Theorem 1631 (*$^{\text{Lag}}\hat{\mathbb{P}}_k^d$ reference Lagrange finite element*).

**Definition 1437 (reference simplex)**

is explicitly cited in the proof of:

Lemma 1438 (*coordinates in reference simplex are smaller than 1*),

Lemma 1439 (*nontrivial reference simplex*),

Lemma 1443 (*simplex of reference vertices is reference simplex*),

Lemma 1452 (*reference simplex is nontrivial in $\mathbb{R}$*),

Lemma 1550 (*properties of geometric mapping*),

Lemma 1581 (*properties of geometric $l$-face mapping*).

**Lemma 1438 (coordinates in reference simplex are smaller than 1)**

is not yet used.

**Lemma 1439 (nontrivial reference simplex)**

is explicitly cited in the proof of:

Lemma 1452 (*reference simplex is nontrivial in $\mathbb{R}$*),

Lemma 1557 (*nontrivial simplex*).

**Definition 1440 (simplex)**

is explicitly cited in the proof of:

Lemma 1442 (*coordinates in simplex are smaller than 1*),

Lemma 1443 (*simplex of reference vertices is reference simplex*),

Lemma 1463 (*current simplex is nontrivial in $\mathbb{R}$*),

Lemma 1550 (*properties of geometric mapping*),

Lemma 1574 (*$d$-face is simplex*).

**Lemma 1442 (coordinates in simplex are smaller than 1)**

is not yet used.

**Lemma 1443 (simplex of reference vertices is reference simplex)**

is explicitly cited in the proof of:

Theorem 1631 (*$^{\text{Lag}}\hat{\mathbb{P}}_k^d$ reference Lagrange finite element*).

**Definition 1445 (multi-indices $A_k^1$)**

is explicitly cited in the proof of:

Lemma 1446 (*cardinal of $A_k^1$*),

Lemma 1487 (*multi-indices $A_k^d$ for $d = 1$ is $A_k^1$*).
**Lemma 1446 (cardinal of $A_k^1$)**

is not yet used.

**Definition 1448 (Lagrange polynomials of $\mathbb{P}_k^1$)**

is explicitly cited in the proof of:

Lemma 1449 (Lagrange polynomials is basis of $\mathbb{P}_k^1$),

Lemma 1470 (Lagrange polynomials of $\mathbb{P}_k^1$ are images of reference),

Lemma 1542 (reference Lagrange polynomials of $\mathbb{P}_1^d$ for $d = 1$ are reference Lagrange polynomials of $\mathbb{P}_k^1$ for $k = 1$).

**Lemma 1449 (Lagrange polynomials is basis of $\mathbb{P}_k^1$)**

is explicitly cited in the proof of:

Lemma 1450 (decomposition of $\mathbb{P}_k^1$ polynomial in Lagrange basis),

Lemma 1455 (reference Lagrange basis of $\mathbb{P}_k^1$),

Lemma 1469 (Lagrange basis of $\mathbb{P}_k^1$).

**Lemma 1450 (decomposition of $\mathbb{P}_k^1$ polynomial in Lagrange basis)**

is explicitly cited in the proof of:

Lemma 1458 (reference Lagrange linear forms for $\mathbb{P}_k^1$ are injective),

Lemma 1476 (Lagrange linear forms for $\mathbb{P}_k^1$ are injective).

**Lemma 1452 (reference simplex is nontrivial in $\mathbb{R}$)**

is explicitly cited in the proof of:

Theorem 1460 (Log $\hat{\mathbb{P}}_k^1$ reference Lagrange finite element),

Lemma 1464 (current simplex is image of reference in $\mathbb{R}$).

**Definition 1453 (reference Lagrange nodes of $\mathbb{P}_k^1$)**

is explicitly cited in the proof of:

Lemma 1454 (reference Lagrange nodes are distinct),

Lemma 1462 (properties of geometric mapping in dimension 1),

Lemma 1468 (Lagrange nodes of $\mathbb{P}_k^1$ are images of reference),

Lemma 1542 (reference Lagrange polynomials of $\mathbb{P}_1^d$ for $d = 1$ are reference Lagrange polynomials of $\mathbb{P}_k^1$ for $k = 1$),

Lemma 1600 (reference Lagrange nodes of $\mathbb{P}_k^d$ for $d = 1$ are reference Lagrange nodes of $\mathbb{P}_k^1$).

**Lemma 1454 (reference Lagrange nodes are distinct)**

is explicitly cited in the proof of:

Lemma 1455 (reference Lagrange basis of $\mathbb{P}_k^1$),

Lemma 1458 (reference Lagrange linear forms for $\mathbb{P}_k^1$ are injective),

Lemma 1459 (unisolvence of $\mathbb{P}_k^1$ (reference)).

**Lemma 1455 (reference Lagrange basis of $\mathbb{P}_k^1$)**

is explicitly cited in the proof of:

Lemma 1462 (properties of geometric mapping in dimension 1),

Lemma 1470 (Lagrange polynomials of $\mathbb{P}_k^1$ are images of reference),

Lemma 1471 (geometric mapping of $\mathbb{P}_k^1$ is $\mathbb{P}_k^1$),

Lemma 1542 (reference Lagrange polynomials of $\mathbb{P}_1^d$ for $d = 1$ are reference Lagrange polynomials of $\mathbb{P}_k^1$ for $k = 1$).

**Definition 1456 (reference Lagrange linear forms for $\mathbb{P}_k^1$)**

is explicitly cited in the proof of:

Lemma 1457 (reference Lagrange linear forms for $\mathbb{P}_k^1$ are linear),

Lemma 1458 (reference Lagrange linear forms for $\mathbb{P}_k^1$ are injective),

Lemma 1474 ($\mathbb{P}_k^1$ Lagrange linear forms are images of reference),

Lemma 1613 (reference Lagrange linear forms for $\mathbb{P}_k^d$ for $d = 1$ are reference Lagrange linear forms for $\mathbb{P}_k^1$).
**Lemma 1457 (reference Lagrange linear forms for $\mathbb{P}_k^1$ are linear)**
is not yet used.

**Lemma 1458 (reference Lagrange linear forms for $\mathbb{P}_k^1$ are injective)**
is explicitly cited in the proof of:
Lemma 1459 (unisolvence of $\mathbb{P}_k^1$ (reference)).

**Lemma 1459 (unisolvence of $\mathbb{P}_k^1$ (reference))**
is explicitly cited in the proof of:
Theorem 1460 ($^{\text{Log}}\hat{\mathbb{P}}_k^1$ reference Lagrange finite element).

**Theorem 1460 ($^{\text{Log}}\hat{\mathbb{P}}_k^1$ reference Lagrange finite element)**
is explicitly cited in the proof of:
Lemma 1632 ($^{\text{Log}}\hat{\mathbb{P}}_k^d$ reference Lagrange finite element for $d = 1$ is Lagrange finite element on the reference segment).

**Definition 1461 (geometric mapping in dimension 1)**
is explicitly cited in the proof of:
Lemma 1462 (properties of geometric mapping in dimension 1),
Lemma 1464 (current simplex is image of reference in $\mathbb{R}$),
Lemma 1468 (Lagrange nodes of $\mathbb{P}_k^1$ are images of reference),
Lemma 1470 (Lagrange polynomials of $\mathbb{P}_k^1$ are images of reference),
Lemma 1471 (geometric mapping of $\mathbb{P}_k^1$ is $\mathbb{P}_k^1$),
Lemma 1548 (geometric mapping for $d = 1$ is geometric mapping in dimension 1).

**Lemma 1462 (properties of geometric mapping in dimension 1)**
is not yet used.

**Lemma 1463 (current simplex is nontrivial in $\mathbb{R}$)**
is explicitly cited in the proof of:
Lemma 1464 (current simplex is image of reference in $\mathbb{R}$),
Theorem 1478 ($^{\text{Log}}\mathbb{P}_k^1$ Lagrange finite element).

**Lemma 1464 (current simplex is image of reference in $\mathbb{R}$)**
is not yet used.

**Definition 1465 (Lagrange nodes of $\mathbb{P}_k^1$)**
is explicitly cited in the proof of:
Lemma 1467 (Lagrange nodes are distinct),
Lemma 1468 (Lagrange nodes of $\mathbb{P}_k^1$ are images of reference),
Lemma 1589 (Lagrange nodes of $\mathbb{P}_k^d$ for $d = 1$ are Lagrange nodes of $\mathbb{P}_k^1$).

**Lemma 1467 (Lagrange nodes are distinct)**
is explicitly cited in the proof of:
Lemma 1469 (Lagrange basis of $\mathbb{P}_k^1$),
Lemma 1476 (Lagrange linear forms for $\mathbb{P}_k^1$ are injective),
Lemma 1477 (unisolvence of $\mathbb{P}_k^1$).

**Lemma 1468 (Lagrange nodes of $\mathbb{P}_k^1$ are images of reference)**
is explicitly cited in the proof of:
Lemma 1470 (Lagrange polynomials of $\mathbb{P}_k^1$ are images of reference),
Lemma 1474 ($\mathbb{P}_k^1$ Lagrange linear forms are images of reference).

**Lemma 1469 (Lagrange basis of $\mathbb{P}_k^1$)**
is explicitly cited in the proof of:
Lemma 1470 (Lagrange polynomials of $\mathbb{P}_k^1$ are images of reference),
Lemma 1471 (geometric mapping of $\mathbb{P}_k^1$ is $\mathbb{P}_k^1$).
**Lemma 1470 (Lagrange polynomials of $\mathbb{P}_k^1$ are images of reference)**

is explicitly cited in the proof of:

Lemma 1471 (*geometric mapping of P_k^1 is P_k^1*).

**Lemma 1471 (geometric mapping of $\mathbb{P}_k^1$ is $\mathbb{P}_k^1$)**

is not yet used.

**Definition 1472 (Lagrange linear forms for $\mathbb{P}_k^1$)**

is explicitly cited in the proof of:

Lemma 1473 ($\mathbb{P}_k^1$ Lagrange linear forms are linear),

Lemma 1474 ($\mathbb{P}_k^1$ Lagrange linear forms are images of reference),

Lemma 1476 (Lagrange linear forms for $\mathbb{P}_k^1$ are injective),

Lemma 1609 (Lagrange linear forms for $\mathbb{P}_k^d$ for $d=1$ are Lagrange linear forms for $\mathbb{P}_k^1$).

**Lemma 1473 ($\mathbb{P}_k^1$ Lagrange linear forms are linear)**

is not yet used.

**Lemma 1474 ($\mathbb{P}_k^1$ Lagrange linear forms are images of reference)**

is not yet used.

**Lemma 1476 (Lagrange linear forms for $\mathbb{P}_k^1$ are injective)**

is explicitly cited in the proof of:

Lemma 1477 (unisolvence of $\mathbb{P}_k^1$),

Lemma 1625 (Lagrange linear forms for $\mathbb{P}_k^d$ are injective).

**Lemma 1477 (unisolvence of $\mathbb{P}_k^1$)**

is explicitly cited in the proof of:

Theorem 1478 ($^{\text{Log}}\mathbb{P}_k^1$ Lagrange finite element).

**Theorem 1478 ($^{\text{Log}}\mathbb{P}_k^1$ Lagrange finite element)**

is explicitly cited in the proof of:

Lemma 1630 ($^{\text{Log}}\mathbb{P}_k^d$ Lagrange finite element for $d = 1$ is Lagrange finite element on a current segment).

**Definition 1480 (length of multi-indices)**

is explicitly cited in the proof of:

Lemma 1481 (length of multi-indices is additive),

Lemma 1487 (multi-indices $A_k^d$ for $d = 1$ is $A_k^1$),

Lemma 1488 (indices are smaller than maximal length),

Lemma 1500 (cardinal of $C_k^d$ and $A_k^{d-1}$),

Lemma 1501 (cardinal of $A_{k,i}^d$ and $A_k^{d-1}$),

Lemma 1539 (affine mapping of monomials is $\mathbb{P}_k^l$),

Lemma 1591 (barycentric coordinates of Lagrange nodes of $\mathbb{P}_k^d$),

Lemma 1592 (vertices are Lagrange nodes of $\mathbb{P}_k^d$),

Lemma 1595 (equivalent definition of sub-vertices of Lagrange nodes of $\mathbb{P}_k^d$),

Lemma 1598 ($\mathbb{P}_{k-1}^d$ sub-nodes of sub-vertices are some nodes of $\mathbb{P}_{k-1}^d$).

**Lemma 1481 (length of multi-indices is additive)**

is explicitly cited in the proof of:

Lemma 1514 (product of monomials).

**Definition 1482 (factorial of multi-indices)**

is explicitly cited in the proof of:

Lemma 1483 (factorial of multi-index is positive),

Lemma 1522 (partial derivative of monomials at 0).
**Lemma 1483 (factorial of multi-index is positive)**

is explicitly cited in the proof of:

Lemma 1523 (*monomials are free in P<sup>d</sup><sub>k</sub>*)

**Definition 1484 (Kronecker delta of multi-indices)**

is explicitly cited in the proof of:

Lemma 1485 (*value of Kronecker delta of multi-indices*),

Lemma 1522 (*partial derivative of monomials at 0*).

**Lemma 1485 (value of Kronecker delta of multi-indices)**

is not yet used.

**Definition 1486 (sets of multi-indices $A_k^d$ and $C_k^d$)**

is explicitly cited in the proof of:

Lemma 1487 (*multi-indices $A_k^d$ for $d = 1$ is $A_k^1$*),

Lemma 1488 (*indices are smaller than maximal length*),

Lemma 1489 (*first $C_k^d$*),

Lemma 1493 (*slices of $C_k^d$*),

Lemma 1496 (*$C_k^d$ are layers of $A_k^d$*),

Lemma 1500 (*cardinal of $C_k^d$ and $A_k^{d-1}$*),

Lemma 1501 (*cardinal of $A_{k,i}^d$ and $A_k^{d-1}$*),

Lemma 1512 (*monomials of $C_k^d$ have degree $k$*),

Lemma 1515 (*product of monomial and polynomial*),

Lemma 1521 (*derivating more than degree is 0*),

Lemma 1539 (*affine mapping of monomials is $\mathbb{P}_k^l$*),

Lemma 1591 (*barycentric coordinates of Lagrange nodes of $\mathbb{P}_k^d$*),

Lemma 1592 (*vertices are Lagrange nodes of $\mathbb{P}_k^d$*),

Lemma 1595 (*equivalent definition of sub-vertices of Lagrange nodes of $\mathbb{P}_k^d$*),

Lemma 1605 (*face hyperplanes of Lagrange nodes of $\mathbb{P}_k^d$*),

Lemma 1607 (*image of nodes by geometric hyperface mapping*).

**Lemma 1487 (multi-indices $A_k^d$ for $d = 1$ is $A_k^1$)**

is not yet used.

**Lemma 1488 (indices are smaller than maximal length)**

is explicitly cited in the proof of:

Lemma 1489 (*first $C_k^d$*),

Lemma 1493 (*slices of $C_k^d$*),

Lemma 1591 (*barycentric coordinates of Lagrange nodes of $\mathbb{P}_k^d$*).

**Lemma 1489 (first $C_k^d$)**

is explicitly cited in the proof of:

Lemma 1495 (*cardinal of $C_k^d$*),

Lemma 1497 (*first multi-indices $A_k^d$*),

Lemma 1504 (*monomial in $d$ variables for $d = 1$ is monomial of a single variable*).

**Definition 1491 (slices $\tilde{S}_{k,i}^d$ and $\tilde{S}_{k,i}^d$)**

is explicitly cited in the proof of:

Lemma 1493 (*slices of $C_k^d$*),

Lemma 1494 (*cardinal of slices of $C_k^d$*),

Lemma 1529 (*decomposition of $\mathbb{P}_k^d$*).

**Lemma 1493 (slices of $C_k^d$)**

is explicitly cited in the proof of:

Lemma 1495 (*cardinal of $C_k^d$*),

Lemma 1529 (*decomposition of $\mathbb{P}_k^d$*).
**Lemma 1494 (cardinal of slices of $C_k^d$)**

is explicitly cited in the proof of:

Lemma 1495 (cardinal of $C_k^d$),

Lemma 1529 (decomposition of $\mathbb{P}_k^d$).

**Lemma 1495 (cardinal of $C_k^d$)**

is explicitly cited in the proof of:

Lemma 1498 (cardinal of $\mathcal{A}_k^d$),

Lemma 1500 (cardinal of $C_k^d$ and $\mathcal{A}_k^{d-1}$),

Lemma 1605 (face hyperplanes of Lagrange nodes of $\mathbb{P}_k^d$).

**Lemma 1496 ($C_k^d$ are layers of $\mathcal{A}_k^d$)**

is explicitly cited in the proof of:

Lemma 1497 (first multi-indices $\mathcal{A}_k^d$),

Lemma 1498 (cardinal of $\mathcal{A}_k^d$),

Lemma 1508 ($\mathbb{P}_k^d$ is nondecreasing sequence in $k$),

Lemma 1513 ($\mathbb{P}_k^d$ is space of degree at most $k$),

Lemma 1529 (decomposition of $\mathbb{P}_k^d$),

Lemma 1540 (affine mapping of $\mathbb{P}_k^d$ is $\mathbb{P}_k^l$),

Lemma 1598 ($\mathbb{P}_{k-1}^d$ sub-nodes of sub-vertices are some nodes of $\mathbb{P}_{k-1}^d$),

Lemma 1625 (Lagrange linear forms for $\mathbb{P}_k^d$ are injective).

**Lemma 1497 (first multi-indices $\mathcal{A}_k^d$)**

is explicitly cited in the proof of:

Lemma 1506 (polynomial space $\mathbb{P}_k^d$ for $d=1$ is $\mathbb{P}_k^1$),

Lemma 1509 (constant and affine spaces $\mathbb{P}_0^d$ and $\mathbb{P}_1^d$),

Lemma 1589 (Lagrange nodes of $\mathbb{P}_k^d$ for $d=1$ are Lagrange nodes of $\mathbb{P}_k^1$),

Lemma 1593 (Lagrange nodes of $\mathbb{P}_1^d$ are vertices),

Lemma 1600 (reference Lagrange nodes of $\mathbb{P}_k^d$ for $d=1$ are reference Lagrange nodes of $\mathbb{P}_k^1$),

Lemma 1609 (Lagrange linear forms for $\mathbb{P}_k^d$ for $d=1$ are Lagrange linear forms for $\mathbb{P}_k^1$),

Lemma 1613 (reference Lagrange linear forms for $\mathbb{P}_k^d$ for $d=1$ are reference Lagrange linear forms for $\mathbb{P}_k^1$),

Lemma 1619 (Lagrange linear forms for $\mathbb{P}_1^d$ are injective).

**Lemma 1498 (cardinal of $\mathcal{A}_k^d$)**

is explicitly cited in the proof of:

Lemma 1500 (cardinal of $C_k^d$ and $\mathcal{A}_k^{d-1}$),

Lemma 1525 (dimension of $\mathbb{P}_k^d$),

Lemma 1590 (number of Lagrange nodes of $\mathbb{P}_k^d$),

Lemma 1602 (number of reference Lagrange nodes of $\mathbb{P}_k^d$),

Lemma 1605 (face hyperplanes of Lagrange nodes of $\mathbb{P}_k^d$).

**Lemma 1500 (cardinal of $C_k^d$ and $\mathcal{A}_k^{d-1}$)**

is explicitly cited in the proof of:

Lemma 1605 (face hyperplanes of Lagrange nodes of $\mathbb{P}_k^d$),

Lemma 1607 (image of nodes by geometric hyperface mapping),

Lemma 1625 (Lagrange linear forms for $\mathbb{P}_k^d$ are injective),

Lemma 1628 (face unisolvence of $\mathbb{P}_k^d$).

**Lemma 1501 (cardinal of $\mathcal{A}_{k,i}^d$ and $\mathcal{A}_k^{d-1}$)**

is explicitly cited in the proof of:

Lemma 1605 (face hyperplanes of Lagrange nodes of $\mathbb{P}_k^d$),

Lemma 1607 (image of nodes by geometric hyperface mapping),

Lemma 1628 (face unisolvence of $\mathbb{P}_k^d$).

**Definition 1503 (monomial in $d$ variables)**

is explicitly cited in the proof of:
Lemma 1504 (monomial in $d$ variables for $d = 1$ is monomial of a single variable),

Lemma 1512 (monomials of $C_k^d$ have degree $k$),

Lemma 1514 (product of monomials),

Lemma 1517 (partial derivative of monomials),

Lemma 1539 (affine mapping of monomials is $\mathbb{P}_k^l$).

**Lemma 1504 (monomial in $d$ variables for $d = 1$ is monomial of a single variable)**

is not yet used.

**Definition 1505 (polynomial space $\mathbb{P}_k^d$)**

is explicitly cited in the proof of:

Lemma 1506 (polynomial space $\mathbb{P}_k^d$ for $d = 1$ is $\mathbb{P}_k^1$),

Lemma 1507 ($\mathbb{P}_k^d$ is vector space),

Lemma 1508 ($\mathbb{P}_k^d$ is nondecreasing sequence in $k$),

Lemma 1509 (constant and affine spaces $\mathbb{P}_0^d$ and $\mathbb{P}_1^d$),

Lemma 1513 ($\mathbb{P}_k^d$ is space of degree at most $k$),

Lemma 1515 (product of monomial and polynomial),

Lemma 1516 (product of two polynomials),

Lemma 1519 (partial derivative is linear),

Lemma 1523 (monomials are free in $\mathbb{P}_k^d$),

Lemma 1524 (monomials are a basis of $\mathbb{P}_k^d$),

Lemma 1529 (decomposition of $\mathbb{P}_k^d$),

Lemma 1533 ($\mathbb{P}_k^d$ is nondecreasing sequence in $d$),

Lemma 1540 (affine mapping of $\mathbb{P}_k^d$ is $\mathbb{P}_k^l$).

**Lemma 1506 (polynomial space $\mathbb{P}_k^d$ for $d = 1$ is $\mathbb{P}_k^1$)**

is not yet used.

**Lemma 1507 ($\mathbb{P}_k^d$ is vector space)**

is explicitly cited in the proof of:

Lemma 1516 (product of two polynomials),

Lemma 1529 (decomposition of $\mathbb{P}_k^d$),

Lemma 1531 (isomorphism between $\mathbb{P}_k^d$ and $\mathbb{P}_k^{d-1} \times \mathbb{P}_k^{d-1}$),

Lemma 1533 ($\mathbb{P}_k^d$ is nondecreasing sequence in $d$),

Lemma 1537 (product of two polynomials (alternate proof)),

Lemma 1540 (affine mapping of $\mathbb{P}_k^d$ is $\mathbb{P}_k^l$).

**Lemma 1508 ($\mathbb{P}_k^d$ is nondecreasing sequence in $k$)**

is explicitly cited in the proof of:

Lemma 1529 (decomposition of $\mathbb{P}_k^d$),

Lemma 1537 (product of two polynomials (alternate proof)),

Lemma 1540 (affine mapping of $\mathbb{P}_k^d$ is $\mathbb{P}_k^l$).

**Lemma 1509 (constant and affine spaces $\mathbb{P}_0^d$ and $\mathbb{P}_1^d$)**

is explicitly cited in the proof of:

Lemma 1526 (isomorphism between $\mathbb{P}_0^d$ and $\mathbb{P}_0^{d-1}$),

Lemma 1529 (decomposition of $\mathbb{P}_k^d$),

Lemma 1533 ($\mathbb{P}_k^d$ is nondecreasing sequence in $d$),

Lemma 1537 (product of two polynomials (alternate proof)),

Lemma 1539 (affine mapping of monomials is $\mathbb{P}_k^l$),

Lemma 1543 (reference Lagrange polynomials is basis of $\mathbb{P}_1^d$),

Lemma 1545 (differential of reference Lagrange polynomials),

Lemma 1554 (Lagrange polynomials is basis of $\mathbb{P}_1^d$),

Lemma 1556 (differential of Lagrange polynomials $\mathbb{P}_1^d$),

Lemma 1615 (Lagrange linear forms for $\mathbb{P}_0^d$ are injective).
**Definition 1510 (degree of polynomial)**

is explicitly cited in the proof of:

Lemma 1511 (*values of degree of polynomial*),

Lemma 1512 (*monomials of $C_k^d$ have degree $k$)*,

Lemma 1513 (*$\mathbb{P}_k^d$ is space of degree at most $k$)*,

Lemma 1543 (*reference Lagrange polynomials is basis of $\mathbb{P}_1^d$*).

**Lemma 1511 (values of degree of polynomial)**

is not yet used.

**Lemma 1512 (monomials of $C_k^d$ have degree $k$)**

is explicitly cited in the proof of:

Lemma 1514 (*product of monomials*),

Lemma 1517 (*partial derivative of monomials*),

Lemma 1529 (*decomposition of $\mathbb{P}_k^d$*).

**Lemma 1513 ($\mathbb{P}_k^d$ is space of degree at most $k$)**

is not yet used.

**Lemma 1514 (product of monomials)**

is explicitly cited in the proof of:

Lemma 1515 (*product of monomial and polynomial*).

**Lemma 1515 (product of monomial and polynomial)**

is explicitly cited in the proof of:

Lemma 1516 (*product of two polynomials*),

Lemma 1537 (*product of two polynomials (alternate proof)*).

**Lemma 1516 (product of two polynomials)**

is explicitly cited in the proof of:

Lemma 1538 (*product of polynomials*).

**Lemma 1517 (partial derivative of monomials)**

is explicitly cited in the proof of:

Lemma 1519 (*partial derivative is linear*),

Lemma 1521 (*derivating more than degree is 0*),

Lemma 1522 (*partial derivative of monomials at 0*).

**Lemma 1519 (partial derivative is linear)**

is explicitly cited in the proof of:

Lemma 1520 (*partial derivative of 0*),

Lemma 1523 (*monomials are free in $\mathbb{P}_k^d$*).

**Lemma 1520 (partial derivative of 0)**

is explicitly cited in the proof of:

Lemma 1523 (*monomials are free in $\mathbb{P}_k^d$*).

**Lemma 1521 (derivating more than degree is 0)**

is not yet used.

**Lemma 1522 (partial derivative of monomials at 0)**

is explicitly cited in the proof of:

Lemma 1523 (*monomials are free in $\mathbb{P}_k^d$*).

**Lemma 1523 (monomials are free in $\mathbb{P}_k^d$)**

is explicitly cited in the proof of:

Lemma 1524 (*monomials are a basis of $\mathbb{P}_k^d$*),

Lemma 1529 (*decomposition of $\mathbb{P}_k^d$*).
**Lemma 1524 (monomials are a basis of $\mathbb{P}_k^d$)**

is explicitly cited in the proof of:

Lemma 1525 (*dimension of P_k^d*).

**Lemma 1525 (dimension of $\mathbb{P}_k^d$)**

is explicitly cited in the proof of:

Lemma 1526 (*isomorphism between $\mathbb{P}_0^d$ and $\mathbb{P}_0^{d-1}$*),

Lemma 1531 (*isomorphism between $\mathbb{P}_k^d$ and $\mathbb{P}_k^{d-1} \times \mathbb{P}_{k-1}^d$*),

Lemma 1543 (*reference Lagrange polynomials is basis of $\mathbb{P}_1^d$*),

Lemma 1554 (*Lagrange polynomials is basis of $\mathbb{P}_1^d$*),

Lemma 1617 (*unisolvence of $\mathbb{P}_0^d$*),

Lemma 1620 (*unisolvence of $\mathbb{P}_1^d$*),

Theorem 1626 (*unisolvence of $\mathbb{P}_k^d$*),

Theorem 1629 (*$\text{Log}\mathbb{P}_k^d$ Lagrange finite element*).

**Lemma 1526 (isomorphism between $\mathbb{P}_0^d$ and $\mathbb{P}_0^{d-1}$)**

is explicitly cited in the proof of:

Lemma 1534 (*expression of $\mathbb{P}_k^d$ as polynomial of $x_d$*).

**Lemma 1529 (decomposition of $\mathbb{P}_k^d$)**

is explicitly cited in the proof of:

Lemma 1531 (*isomorphism between $\mathbb{P}_k^d$ and $\mathbb{P}_k^{d-1} \times \mathbb{P}_{k-1}^d$*),

Lemma 1534 (*expression of $\mathbb{P}_k^d$ as polynomial of $x_d$*),

Lemma 1537 (*product of two polynomials (alternate proof)*),

Lemma 1621 (*factorization of zero polynomial on last reference hyperplane*).

**Lemma 1531 (isomorphism between $\mathbb{P}_k^d$ and $\mathbb{P}_k^{d-1} \times \mathbb{P}_{k-1}^d$)**

is explicitly cited in the proof of:

Lemma 1533 (*$\mathbb{P}_k^d$ is nondecreasing sequence in $d$*).

**Lemma 1533 ($\mathbb{P}_k^d$ is nondecreasing sequence in $d$)**

is explicitly cited in the proof of:

Lemma 1537 (*product of two polynomials (alternate proof)*).

**Lemma 1534 (expression of $\mathbb{P}_k^d$ as polynomial of $x_d$)**

is not yet used.

**Lemma 1537 (product of two polynomials (alternate proof))**

is explicitly cited in the proof of:

Lemma 1538 (*product of polynomials*).

**Lemma 1538 (product of polynomials)**

is explicitly cited in the proof of:

Lemma 1539 (*affine mapping of monomials is $\mathbb{P}_k^l$*).

**Lemma 1539 (affine mapping of monomials is $\mathbb{P}_k^l$)**

is explicitly cited in the proof of:

Lemma 1540 (*affine mapping of $\mathbb{P}_k^d$ is $\mathbb{P}_k^l$*).

**Lemma 1540 (affine mapping of $\mathbb{P}_k^d$ is $\mathbb{P}_k^l$)**

is explicitly cited in the proof of:

Lemma 1582 (*geometric l-face mapping of $\mathbb{P}_k^d$ is $\mathbb{P}_k^l$*),

Lemma 1583 (*geometric mapping of $\mathbb{P}_k^d$ is $\mathbb{P}_k^l$*),

Lemma 1623 (*factorization of zero polynomial on hyperplane $\mathbb{P}_k^d$*).

**Definition 1541 (reference Lagrange polynomials of $\mathbb{P}_1^d$)**

is explicitly cited in the proof of:
Lemma 1542 (reference Lagrange polynomials of $\mathbb{P}_1^d$ for $d = 1$ are reference Lagrange polynomials of $\mathbb{P}_k^1$ for $k = 1$),

Lemma 1543 (reference Lagrange polynomials is basis of $\mathbb{P}_1^d$),

Lemma 1548 (geometric mapping for $d = 1$ is geometric mapping in dimension 1),

Lemma 1549 (reference geometric mapping is identity),

Lemma 1550 (properties of geometric mapping),

Lemma 1564 (reference face hyperplane),

Lemma 1565 (face hyperplane is image of reference face hyperplane),

Lemma 1581 (properties of geometric l-face mapping),

Lemma 1621 (factorization of zero polynomial on last reference hyperplane).

**Lemma 1542 (reference Lagrange polynomials of $\mathbb{P}_1^d$ for $d = 1$ are reference Lagrange polynomials of $\mathbb{P}_k^1$ for $k = 1$)**

is not yet used.

**Lemma 1543 (reference Lagrange polynomials is basis of $\mathbb{P}_1^d$)**

is explicitly cited in the proof of:

Lemma 1545 (differential of reference Lagrange polynomials),

Lemma 1550 (properties of geometric mapping),

Lemma 1554 (Lagrange polynomials is basis of $\mathbb{P}_1^d$),

Lemma 1581 (properties of geometric l-face mapping),

Lemma 1586 (geometric mapping with permutation).

**Lemma 1545 (differential of reference Lagrange polynomials)**

is explicitly cited in the proof of:

Lemma 1556 (differential of Lagrange polynomials $\mathbb{P}_1^d$).

**Definition 1547 (geometric mapping)**

is explicitly cited in the proof of:

Lemma 1548 (geometric mapping for $d = 1$ is geometric mapping in dimension 1),

Lemma 1549 (reference geometric mapping is identity),

Lemma 1550 (properties of geometric mapping),

Lemma 1560 (Lagrange polynomials of $\mathbb{P}_1^d$ are barycentric coordinate),

Lemma 1565 (face hyperplane is image of reference face hyperplane),

Lemma 1579 (geometric d-face mapping is geometric mapping).

**Lemma 1548 (geometric mapping for $d = 1$ is geometric mapping in dimension 1)**

is not yet used.

**Lemma 1549 (reference geometric mapping is identity)**

is explicitly cited in the proof of:

Lemma 1553 (Lagrange polynomials of reference vertices are reference Lagrange polynomials of $\mathbb{P}_1^d$).

**Lemma 1550 (properties of geometric mapping)**

is explicitly cited in the proof of:

Lemma 1551 (differential of geometric mapping),

Lemma 1552 (Lagrange polynomials of $\mathbb{P}_1^d$),

Lemma 1554 (Lagrange polynomials is basis of $\mathbb{P}_1^d$),

Lemma 1557 (nontrivial simplex),

Lemma 1560 (Lagrange polynomials of $\mathbb{P}_1^d$ are barycentric coordinate),

Lemma 1583 (geometric mapping of $\mathbb{P}_k^d$ is $\mathbb{P}_k^d$),

Lemma 1604 (Lagrange nodes of $\mathbb{P}_k^d$ are image of reference).

**Lemma 1551 (differential of geometric mapping)**

is explicitly cited in the proof of:
Lemma 1556 (differential of Lagrange polynomials $\mathbb{P}_1^d$),
Lemma 1557 (nontrivial simplex).

**Lemma 1552 (Lagrange polynomials of $\mathbb{P}_1^d$)**

is explicitly cited in the proof of:

Lemma 1553 (Lagrange polynomials of reference vertices are reference Lagrange polynomials of $\mathbb{P}_1^d$),

Lemma 1554 (Lagrange polynomials is basis of $\mathbb{P}_1^d$),

Lemma 1560 (Lagrange polynomials of $\mathbb{P}_1^d$ are barycentric coordinate).

**Lemma 1553 (Lagrange polynomials of reference vertices are reference Lagrange polynomials of $\mathbb{P}_1^d$)**

is explicitly cited in the proof of:

Lemma 1564 (reference face hyperplane).

**Lemma 1554 (Lagrange polynomials is basis of $\mathbb{P}_1^d$)**

is explicitly cited in the proof of:

Lemma 1555 (decomposition of $\mathbb{P}_1^d$ polynomial in Lagrange basis),

Lemma 1556 (differential of Lagrange polynomials $\mathbb{P}_1^d$).

**Lemma 1555 (decomposition of $\mathbb{P}_1^d$ polynomial in Lagrange basis)**

is explicitly cited in the proof of:

Lemma 1561 (decomposition of $\mathbb{P}_1^d$ polynomial with barycentric coordinates),

Lemma 1618 (decomposition of $\mathbb{P}_1^d$ polynomial with $\sigma_\alpha$).

**Lemma 1556 (differential of Lagrange polynomials $\mathbb{P}_1^d$)**

is not yet used.

**Lemma 1557 (nontrivial simplex)**

is explicitly cited in the proof of:

Theorem 1629 ($\mathrm{Log}\mathbb{P}_k^d$ Lagrange finite element).

**Lemma 1559 (barycentric coordinate)**

is explicitly cited in the proof of:

Lemma 1560 (Lagrange polynomials of $\mathbb{P}_1^d$ are barycentric coordinate),

Lemma 1563 (equivalent definition of face hyperplane),

Lemma 1586 (geometric mapping with permutation),

Lemma 1590 (number of Lagrange nodes of $\mathbb{P}_k^d$),

Lemma 1591 (barycentric coordinates of Lagrange nodes of $\mathbb{P}_k^d$),

Lemma 1602 (number of reference Lagrange nodes of $\mathbb{P}_k^d$).

**Lemma 1560 (Lagrange polynomials of $\mathbb{P}_1^d$ are barycentric coordinate)**

is explicitly cited in the proof of:

Lemma 1561 (decomposition of $\mathbb{P}_1^d$ polynomial with barycentric coordinates),

Lemma 1564 (reference face hyperplane),

Lemma 1586 (geometric mapping with permutation).

**Lemma 1561 (decomposition of $\mathbb{P}_1^d$ polynomial with barycentric coordinates)**

is explicitly cited in the proof of:

Lemma 1618 (decomposition of $\mathbb{P}_1^d$ polynomial with $\sigma_\alpha$).

**Definition 1562 (face hyperplane)**

is explicitly cited in the proof of:

Lemma 1563 (equivalent definition of face hyperplane),

Lemma 1565 (face hyperplane is image of reference face hyperplane).
**Lemma 1563 (equivalent definition of face hyperplane)**

is explicitly cited in the proof of:

Lemma 1564 (reference face hyperplane),

Lemma 1567 (hyperface is included in face hyperplane),

Lemma 1576 ((d − 1)-face is hyperface),

Lemma 1586 (geometric mapping with permutation),

Lemma 1605 (face hyperplanes of Lagrange nodes of $\mathbb{P}_k^d$),

Lemma 1623 (factorization of zero polynomial on hyperplane $\mathbb{P}_k^d$),

Lemma 1625 (Lagrange linear forms for $\mathbb{P}_k^d$ are injective).

**Lemma 1564 (reference face hyperplane)**

is explicitly cited in the proof of:

Lemma 1586 (geometric mapping with permutation),

Lemma 1621 (factorization of zero polynomial on last reference hyperplane).

**Lemma 1565 (face hyperplane is image of reference face hyperplane)**

is not yet used.

**Definition 1566 (hyperface)**

is explicitly cited in the proof of:

Lemma 1567 (hyperface is included in face hyperplane),

Lemma 1576 ((d − 1)-face is hyperface).

**Lemma 1567 (hyperface is included in face hyperplane)**

is not yet used.

**Definition 1568 (l-face affine space)**

is explicitly cited in the proof of:

Lemma 1569 (equivalent definition of l-face affine space),

Lemma 1573 (l-face is included in l-face affine space),

Lemma 1576 ((d − 1)-face is hyperface),

Lemma 1581 (properties of geometric l-face mapping).

**Lemma 1569 (equivalent definition of l-face affine space)**

is explicitly cited in the proof of:

Lemma 1570 (d-face affine space is full space),

Lemma 1571 (0-face affine space is vertex),

Lemma 1581 (properties of geometric l-face mapping).

**Lemma 1570 (d-face affine space is full space)**

is explicitly cited in the proof of:

Lemma 1586 (geometric mapping with permutation).

**Lemma 1571 (0-face affine space is vertex)**

is not yet used.

**Definition 1572 (l-face)**

is explicitly cited in the proof of:

Lemma 1573 (l-face is included in l-face affine space),

Lemma 1574 (d-face is simplex),

Lemma 1576 ((d − 1)-face is hyperface),

Lemma 1581 (properties of geometric l-face mapping).

**Lemma 1573 (l-face is included in l-face affine space)**

is not yet used.
**Lemma 1574 (d-face is simplex)**

is explicitly cited in the proof of:

Lemma 1586 (*geometric mapping with permutation*).

**Lemma 1576 (($d-1$)-face is hyperface)**

is explicitly cited in the proof of:

Lemma 1584 (*geometric hyperface mapping*),

Lemma 1585 (*hyperface geometric mapping of $\mathbb{P}_k^d$ is $\mathbb{P}_k^{d-1}$*).

**Definition 1578 (geometric $l$-face mapping)**

is explicitly cited in the proof of:

Lemma 1579 (*geometric $d$-face mapping is geometric mapping*),

Lemma 1581 (*properties of geometric $l$-face mapping*),

Lemma 1586 (*geometric mapping with permutation*).

**Lemma 1579 (geometric $d$-face mapping is geometric mapping)**

is explicitly cited in the proof of:

Lemma 1583 (*geometric mapping of $\mathbb{P}_k^d$ is $\mathbb{P}_k^{d}$*).

**Lemma 1581 (properties of geometric $l$-face mapping)**

is explicitly cited in the proof of:

Lemma 1582 (*geometric $l$-face mapping of $\mathbb{P}_k^d$ is $\mathbb{P}_k^l$*),

Lemma 1584 (*geometric hyperface mapping*),

Lemma 1586 (*geometric mapping with permutation*).

**Lemma 1582 (geometric $l$-face mapping of $\mathbb{P}_k^d$ is $\mathbb{P}_k^l$)**

is explicitly cited in the proof of:

Lemma 1583 (*geometric mapping of $\mathbb{P}_k^d$ is $\mathbb{P}_k^d$*),

Lemma 1585 (*hyperface geometric mapping of $\mathbb{P}_k^d$ is $\mathbb{P}_k^{d-1}$*).

**Lemma 1583 (geometric mapping of $\mathbb{P}_k^d$ is $\mathbb{P}_k^d$)**

is not yet used.

**Lemma 1584 (geometric hyperface mapping)**

is explicitly cited in the proof of:

Lemma 1607 (*image of nodes by geometric hyperface mapping*),

Lemma 1625 (*Lagrange linear forms for $\mathbb{P}_k^d$ are injective*),

Lemma 1628 (*face unisolvence of $\mathbb{P}_k^d$*).

**Lemma 1585 (hyperface geometric mapping of $\mathbb{P}_k^d$ is $\mathbb{P}_k^{d-1}$)**

is explicitly cited in the proof of:

Lemma 1625 (*Lagrange linear forms for $\mathbb{P}_k^d$ are injective*),

Lemma 1628 (*face unisolvence of $\mathbb{P}_k^d$*).

**Lemma 1586 (geometric mapping with permutation)**

is explicitly cited in the proof of:

Lemma 1623 (*factorization of zero polynomial on hyperplane $\mathbb{P}_k^d$*).

**Definition 1588 (Lagrange nodes of $\mathbb{P}_k^d$)**

is explicitly cited in the proof of:

Lemma 1589 (*Lagrange nodes of $\mathbb{P}_k^d$ for $d = 1$ are Lagrange nodes of $\mathbb{P}_k^1$*),

Lemma 1590 (*number of Lagrange nodes of $\mathbb{P}_k^d$*),

Lemma 1591 (*barycentric coordinates of Lagrange nodes of $\mathbb{P}_k^d$*),

Lemma 1592 (*vertices are Lagrange nodes of $\mathbb{P}_k^d$*),

Lemma 1599 (*reference Lagrange nodes of $\mathbb{P}_k^d$*),

Lemma 1604 (*Lagrange nodes of $\mathbb{P}_k^d$ are image of reference*),

Lemma 1625 (*Lagrange linear forms for $\mathbb{P}_k^d$ are injective*).
**Lemma 1589 (Lagrange nodes of $\mathbb{P}_k^d$ for $d=1$ are Lagrange nodes of $\mathbb{P}_k^1$)**

is explicitly cited in the proof of:

Lemma 1609 (Lagrange linear forms for $\mathbb{P}_k^d$ for $d=1$ are Lagrange linear forms for $\mathbb{P}_k^1$).

**Lemma 1590 (number of Lagrange nodes of $\mathbb{P}_k^d$)**

is explicitly cited in the proof of:

Lemma 1605 (face hyperplanes of Lagrange nodes of $\mathbb{P}_k^d$),

Lemma 1611 (cardinal of Lagrange linear forms of $\mathbb{P}_k^d$).

**Lemma 1591 (barycentric coordinates of Lagrange nodes of $\mathbb{P}_k^d$)**

is explicitly cited in the proof of:

Lemma 1595 (equivalent definition of sub-vertices of Lagrange nodes of $\mathbb{P}_k^d$),

Lemma 1598 ($\mathbb{P}_{k-1}^d$ sub-nodes of sub-vertices are some nodes of $\mathbb{P}_{k-1}^d$),

Lemma 1599 (reference Lagrange nodes of $\mathbb{P}_k^d$),

Lemma 1605 (face hyperplanes of Lagrange nodes of $\mathbb{P}_k^d$),

Lemma 1607 (image of nodes by geometric hyperface mapping).

**Lemma 1592 (vertices are Lagrange nodes of $\mathbb{P}_k^d$)**

is explicitly cited in the proof of:

Lemma 1593 (Lagrange nodes of $\mathbb{P}_1^d$ are vertices).

**Lemma 1593 (Lagrange nodes of $\mathbb{P}_1^d$ are vertices)**

is explicitly cited in the proof of:

Lemma 1618 (decomposition of $\mathbb{P}_1^d$ polynomial with $\sigma_\alpha$).

**Definition 1594 (sub-vertices of Lagrange nodes of $\mathbb{P}_k^d$)**

is explicitly cited in the proof of:

Lemma 1595 (equivalent definition of sub-vertices of Lagrange nodes of $\mathbb{P}_k^d$).

**Lemma 1595 (equivalent definition of sub-vertices of Lagrange nodes of $\mathbb{P}_k^d$)**

is explicitly cited in the proof of:

Lemma 1597 (sub-vertices are affinely independent),

Lemma 1598 ($\mathbb{P}_{k-1}^d$ sub-nodes of sub-vertices are some nodes of $\mathbb{P}_{k-1}^d$).

**Lemma 1597 (sub-vertices are affinely independent)**

is explicitly cited in the proof of:

Lemma 1598 ($\mathbb{P}_{k-1}^d$ sub-nodes of sub-vertices are some nodes of $\mathbb{P}_{k-1}^d$),

Lemma 1625 (Lagrange linear forms for $\mathbb{P}_k^d$ are injective).

**Lemma 1598 ($\mathbb{P}_{k-1}^d$ sub-nodes of sub-vertices are some nodes of $\mathbb{P}_{k-1}^d$)**

is explicitly cited in the proof of:

Lemma 1625 (Lagrange linear forms for $\mathbb{P}_k^d$ are injective).

**Lemma 1599 (reference Lagrange nodes of $\mathbb{P}_k^d$)**

is explicitly cited in the proof of:

Lemma 1600 (reference Lagrange nodes of $\mathbb{P}_k^d$ for $d=1$ are reference Lagrange nodes of $\mathbb{P}_k^1$),

Lemma 1601 (equivalent definition of reference Lagrange nodes of $\mathbb{P}_k^d$),

Lemma 1602 (number of reference Lagrange nodes of $\mathbb{P}_k^d$),

Lemma 1607 (image of nodes by geometric hyperface mapping),

Theorem 1631 (Log $\hat{\mathbb{P}}_k^d$ reference Lagrange finite element).

**Lemma 1600 (reference Lagrange nodes of $\mathbb{P}_k^d$ for $d=1$ are reference Lagrange nodes of $\mathbb{P}_k^1$)**

is explicitly cited in the proof of:

Lemma 1613 (reference Lagrange linear forms for $\mathbb{P}_k^d$ for $d=1$ are reference Lagrange linear forms for $\mathbb{P}_k^1$).
**Lemma 1601 (equivalent definition of reference Lagrange nodes of $\mathbb{P}_k^d$)**
is explicitly cited in the proof of:
Lemma 1604 (Lagrange nodes of $\mathbb{P}_k^d$ are image of reference).

**Lemma 1602 (number of reference Lagrange nodes of $\mathbb{P}_k^d$)**
is not yet used.

**Lemma 1604 (Lagrange nodes of $\mathbb{P}_k^d$ are image of reference)**
is explicitly cited in the proof of:
Lemma 1614 (Lagrange linear forms of $\mathbb{P}_k^d$ are images of reference).

**Lemma 1605 (face hyperplanes of Lagrange nodes of $\mathbb{P}_k^d$)**
is explicitly cited in the proof of:
Lemma 1625 (Lagrange linear forms for $\mathbb{P}_k^d$ are injective),
Lemma 1628 (face unisolvence of $\mathbb{P}_k^d$).

**Lemma 1607 (image of nodes by geometric hyperface mapping)**
is explicitly cited in the proof of:
Lemma 1625 (Lagrange linear forms for $\mathbb{P}_k^d$ are injective),
Lemma 1628 (face unisolvence of $\mathbb{P}_k^d$).

**Definition 1608 (Lagrange linear forms for $\mathbb{P}_k^d$)**
is explicitly cited in the proof of:
Lemma 1609 (Lagrange linear forms for $\mathbb{P}_k^d$ for $d=1$ are Lagrange linear forms for $\mathbb{P}_k^1$),
Lemma 1610 (Lagrange linear forms of $\mathbb{P}_k^d$ are linear),
Lemma 1611 (cardinal of Lagrange linear forms of $\mathbb{P}_k^d$),
Lemma 1614 (Lagrange linear forms of $\mathbb{P}_k^d$ are images of reference),
Lemma 1615 (Lagrange linear forms for $\mathbb{P}_k^0$ are injective),
Lemma 1618 (decomposition of $\mathbb{P}_k^d$ polynomial with $\sigma_\alpha$),
Lemma 1625 (Lagrange linear forms for $\mathbb{P}_k^d$ are injective),
Theorem 1626 (unisolvence of $\mathbb{P}_k^d$),
Lemma 1628 (face unisolvence of $\mathbb{P}_k^d$).

**Lemma 1609 (Lagrange linear forms for $\mathbb{P}_k^d$ for $d=1$ are Lagrange linear forms for $\mathbb{P}_k^1$)**
is explicitly cited in the proof of:
Lemma 1625 (Lagrange linear forms for $\mathbb{P}_k^d$ are injective),
Lemma 1630 ($\text{Log}\mathbb{P}_k^d$ Lagrange finite element for $d=1$ is Lagrange finite element on a current segment).

**Lemma 1610 (Lagrange linear forms of $\mathbb{P}_k^d$ are linear)**
is not yet used.

**Lemma 1611 (cardinal of Lagrange linear forms of $\mathbb{P}_k^d$)**
is explicitly cited in the proof of:
Lemma 1617 (unisolvence of $\mathbb{P}_k^0$),
Lemma 1620 (unisolvence of $\mathbb{P}_k^1$),
Theorem 1626 (unisolvence of $\mathbb{P}_k^d$).

**Definition 1612 (reference Lagrange linear forms for $\mathbb{P}_k^d$)**
is explicitly cited in the proof of:
Lemma 1613 (reference Lagrange linear forms for $\mathbb{P}_k^d$ for $d=1$ are reference Lagrange linear forms for $\mathbb{P}_k^1$),
Lemma 1614 (Lagrange linear forms of $\mathbb{P}_k^d$ are images of reference),
Theorem 1631 ($\text{Log}\hat{\mathbb{P}}_k^d$ reference Lagrange finite element).
**Lemma 1613 (reference Lagrange linear forms for $\mathbb{P}_k^d$ for $d = 1$ are reference Lagrange linear forms for $\mathbb{P}_k^1$)**

is explicitly cited in the proof of:

Lemma 1632 ($^{\text{Log}}\hat{\mathbb{P}}_k^d$ reference Lagrange finite element for $d = 1$ is Lagrange finite element on the reference segment).

**Lemma 1614 (Lagrange linear forms of $\mathbb{P}_k^d$ are images of reference)**

is not yet used.

**Lemma 1615 (Lagrange linear forms for $\mathbb{P}_0^d$ are injective)**

is explicitly cited in the proof of:

Lemma 1617 (unisolvence of $\mathbb{P}_0^d$).

**Lemma 1617 (unisolvence of $\mathbb{P}_0^d$)**

is explicitly cited in the proof of:

Theorem 1629 ($^{\text{Log}}\mathbb{P}_k^d$ Lagrange finite element).

**Lemma 1618 (decomposition of $\mathbb{P}_1^d$ polynomial with $\sigma_\alpha$)**

is explicitly cited in the proof of:

Lemma 1619 (Lagrange linear forms for $\mathbb{P}_1^d$ are injective).

**Lemma 1619 (Lagrange linear forms for $\mathbb{P}_1^d$ are injective)**

is explicitly cited in the proof of:

Lemma 1620 (unisolvence of $\mathbb{P}_1^d$),

Lemma 1625 (Lagrange linear forms for $\mathbb{P}_k^d$ are injective).

**Lemma 1620 (unisolvence of $\mathbb{P}_1^d$)**

is not yet used.

**Lemma 1621 (factorization of zero polynomial on last reference hyperplane)**

is explicitly cited in the proof of:

Lemma 1623 (factorization of zero polynomial on hyperplane $\mathbb{P}_k^d$).

**Lemma 1623 (factorization of zero polynomial on hyperplane $\mathbb{P}_k^d$)**

is explicitly cited in the proof of:

Lemma 1625 (Lagrange linear forms for $\mathbb{P}_k^d$ are injective).

**Lemma 1625 (Lagrange linear forms for $\mathbb{P}_k^d$ are injective)**

is explicitly cited in the proof of:

Theorem 1626 (unisolvence of $\mathbb{P}_k^d$),

Lemma 1628 (face unisolvence of $\mathbb{P}_k^d$).

**Theorem 1626 (unisolvence of $\mathbb{P}_k^d$)**

is explicitly cited in the proof of:

Theorem 1629 ($^{\text{Log}}\mathbb{P}_k^d$ Lagrange finite element).

**Lemma 1628 (face unisolvence of $\mathbb{P}_k^d$)**

is not yet used.

**Theorem 1629 ($^{\text{Log}}\mathbb{P}_k^d$ Lagrange finite element)**

is explicitly cited in the proof of:

Lemma 1630 ($^{\text{Log}}\mathbb{P}_k^d$ Lagrange finite element for $d = 1$ is Lagrange finite element on a current segment),

Theorem 1631 ($^{\text{Log}}\hat{\mathbb{P}}_k^d$ reference Lagrange finite element).

**Lemma 1630 ($^{\text{Log}}\mathbb{P}_k^d$ Lagrange finite element for $d = 1$ is Lagrange finite element on a current segment)**

is not yet used.
**Theorem 1631** ($^{Log}\hat{P}_k^d$ reference Lagrange finite element)

is explicitly cited in the proof of:

Lemma 1632 ($^{Log}\hat{P}_k^d$ reference Lagrange finite element for $d = 1$ is Lagrange finite element on the reference segment).

**Lemma 1632** ($^{Log}\hat{P}_k^d$ reference Lagrange finite element for $d = 1$ is Lagrange finite element on the reference segment)

is not yet used.
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