:: {C}ross-ratio in Real Vector Space
::  by Roland Coghetto

environ

 vocabularies NAT_1, FINSEQ_2, REAL_1, XCMPLX_0, PENCIL_1, ALGSTR_0, ARYTM_1,
      ARYTM_3, CARD_1, EUCLID, FUNCT_1, INCSP_1, NUMBERS, PRE_TOPC, RELAT_1,
      STRUCT_0, SUBSET_1, SUPINF_2, VECTSP_1, XBOOLE_0, TARSKI, FINSEQ_1,
      RLTOPSP1, ANPROJ10, FUNCT_5, RLVECT_1, FUNCOP_1, FUNCT_2, FUNCSDOM,
      BINOP_2, PCOMPS_1, ZFMISC_1;
 notations ORDINAL1, XCMPLX_0, PRE_TOPC, TARSKI, XBOOLE_0, RLVECT_1, XREAL_0,
      SUBSET_1, NUMBERS, FUNCT_1, FUNCT_2, FINSEQ_1, STRUCT_0, ALGSTR_0,
      VECTSP_1, FINSEQ_2, BINOP_1, FUNCOP_1, FINSEQ_4, RVSUM_1, FUNCSDOM,
      BINOP_2, PCOMPS_1, EUCLID, SRINGS_5, RLTOPSP1, EUCLID_4, ZFMISC_1;
 constructors FINSEQ_4, MONOID_0, BINOP_2, FUNCSDOM, PCOMPS_1, SRINGS_5,
      EUCLID_4;
 registrations MEMBERED, ORDINAL1, STRUCT_0, XREAL_0, MONOID_0, EUCLID,
      VALUED_0, XCMPLX_0, NUMBERS, RLTOPSP1, RLVECT_1, RELAT_1, FINSEQ_4;
 requirements REAL, SUBSET, NUMERALS, ARITHM, BOOLE;
 definitions TARSKI, XBOOLE_0;
 equalities STRUCT_0, EUCLID, XCMPLX_0, RLVECT_1, FINSEQ_1, FUNCSDOM, PCOMPS_1,
      FINSEQ_2, RVSUM_1, VALUED_1;
 expansions RLTOPSP1;
 theorems XREAL_0, EUCLID, XCMPLX_1, RVSUM_1, FINSEQ_1, FINSEQ_2, RLTOPSP1,
      EUCLID12, RLVECT_1, FINSEQ_4, JORDAN2B, FUNCSDOM, ALGSTR_0, BINOP_2,
      SRINGS_5, FUNCOP_1, EUCLID_4, ZFMISC_1, CARD_1;

begin :: Preliminaries

registration
  let a,b,c,d be object;
  reduce <*a,b,c,d*>.1 to a;
  reducibility by FINSEQ_4:76;
  reduce <*a,b,c,d*>.2 to b;
  reducibility by FINSEQ_4:76;
  reduce <*a,b,c,d*>.3 to c;
  reducibility by FINSEQ_4:76;
  reduce <*a,b,c,d*>.4 to d;
  reducibility by FINSEQ_4:76;
end;

theorem
  for a,b,c,d,a9,b9,c9,d9 being object st
    <* a, b, c, d *> = <* a9, b9, c9, d9 *> holds
      a = a9 & b = b9 & c = c9 & d = d9
  proof
    let a,b,c,d,a9,b9,c9,d9 be object;
    assume
A1: <* a, b, c, d *> = <* a9, b9, c9, d9 *>;
    set x = <* a,b,c,d *>, y = <*a9,b9,c9,d9 *>;
    x.1 = a & x.2 = b & x.3 = c & x.4 = d & y.1 = a9 & y.2 = b9 &
      y.3 = c9 & y.4 = d9;
    hence thesis by A1;
  end;

definition
  let r be Real;
  attr r is unit means :Def01:
  r = 1;
end;

registration
  cluster non unit for non zero Real;
  existence
  proof
    take 2;
    thus thesis;
  end;
end;

definition
  let r be non unit non zero Real;
  func op1(r) -> non unit non zero Real equals
  1 / r;
  coherence
  proof
    1 / r <> 1
    proof
      assume 1 / r = 1;
      then r * (1 / r) = r;
      hence contradiction by Def01,XCMPLX_1:106;
    end;
    hence thesis by Def01;
  end;
  involutiveness;
end;

definition
  let r being non unit non zero Real;
  func op2(r) -> non unit non zero Real equals
  1 - r;
  coherence
  proof
    r <> 0; then
    1 - r <> 0 & 1 - r <> 1 by Def01;
    hence thesis by Def01;
  end;
  involutiveness;
end;

reserve a,b,r for non unit non zero Real;

theorem Th01:
  op2(op1(r)) = (r - 1) / r &
  op1(op2(r)) = 1 / (1 - r) &
  op1(op2(op1(r))) = r / (r - 1) &
  op2(op1(op2(r))) = r / (r - 1)
  proof
A1: 1 - 1 / r = r / r - 1 / r by XCMPLX_1:60
             .= (r - 1) / r;
    1 - r <> 0 by Def01;
    then 1 - (1 / (1 - r)) = ((1 - r) / (1 - r)) - 1 / (1 - r) by XCMPLX_1:60
                          .= -r / (1 - r)
                          .= r / -(1 - r) by XCMPLX_1:188;
    hence thesis by A1,XCMPLX_1:57;
  end;

theorem
  op2(op1(op2(op1(r)))) = op1(op2(r)) &
  op1(op2(op1(op2(r)))) = op2(op1(r))
  proof
    op2(op1(op2(op1(r)))) = 1 - (r / (r - 1)) &
    op1(op2(op1(op2(r)))) = 1 / (r / (r - 1)) by Th01;
    hence thesis;
  end;

theorem
  op1(a) / op1(b) = b / a;

reserve X for non empty set,
        x for Tuple of 4,X;

theorem Th02:
  4-tuples_on X = the set of all <* d1, d2, d3, d4 *> where
    d1,d2,d3,d4 is Element of X
  proof
    hereby
      let x be object;
      assume x in 4-tuples_on X;
      then consider s be Element of X* such that
A1:   x = s and
A2:   len s = 4;
      1 in Seg 4 & 2 in Seg 4 & 3 in Seg 4 & 4 in Seg 4;
      then 1 in dom s & 2 in dom s & 3 in dom s & 4 in dom s
        by A2,FINSEQ_1:def 3;
      then reconsider d19 = s.1, d29 = s.2, d39 = s.3,
      d49 = s.4 as Element of X by FINSEQ_2:11;
      s = <* d19, d29, d39, d49 *> by A2,FINSEQ_4:76;
      hence x in the set of all <* d1, d2, d3, d4 *>
        where d1,d2,d3,d4 is Element of X by A1;
    end;
    let x be object;
    assume x in the set of all <* d1, d2, d3, d4 *>
      where d1,d2,d3,d4 is Element of X;
    then consider d1, d2, d3, d4 be Element of X such that
A3: x = <* d1, d2, d3,d4 *>;
    len <* d1, d2, d3, d4 *> = 4 & <* d1, d2, d3, d4 *> is Element of X*
      by FINSEQ_1:def 11,FINSEQ_4:76;
    hence x in 4-tuples_on X by A3;
  end;

theorem Th03:
  for a,b,c,d being object st (a = x.1 or a = x.2 or a = x.3 or a = x.4) &
  (b = x.1 or b = x.2 or b = x.3 or b = x.4) &
  (c = x.1 or c = x.2 or c = x.3 or c = x.4) &
  (d = x.1 or d = x.2 or d = x.3 or d = x.4) holds
  <* a,b,c,d *> is Tuple of 4,X
  proof
    let a,b,c,d be object;
    assume
A1: (a = x.1 or a = x.2 or a = x.3 or a = x.4) &
      (b = x.1 or b = x.2 or b = x.3 or b = x.4) &
      (c = x.1 or c = x.2 or c = x.3 or c = x.4) &
      (d = x.1 or d = x.2 or d = x.3 or d = x.4);
      set y = <* a,b,c,d *>;
    dom x = Seg 4 by FINSEQ_2:124; then
A2: 1 in dom x & 2 in dom x & 3 in dom x & 4 in dom x;
    reconsider d19 = y.1, d29 = y.2, d39 = y.3,
      d49 = y.4 as Element of X by A1,A2,FINSEQ_2:11;
    a is Element of X & b is Element of X & c is Element of X &
      d is Element of X by A1,A2,FINSEQ_2:11; then
    y in the set of all <* d1, d2, d3, d4 *> where
      d1,d2,d3,d4 is Element of X;
    then y in 4-tuples_on X by Th02;
    hence thesis by FINSEQ_2:131;
  end;

definition
  let X be non empty set;
  let x be Tuple of 4,X;
  func pi_1342(x) -> Tuple of 4,X equals <* x.1,x.3,x.4,x.2 *>;
  coherence by Th03;
  func pi_1423(x) -> Tuple of 4,X equals <* x.1,x.4,x.2,x.3 *>;
  coherence by Th03;
  func pi_2143(x) -> Tuple of 4,X equals <* x.2,x.1,x.4,x.3 *>;
  coherence by Th03;
  func pi_2314(x) -> Tuple of 4,X equals <* x.2,x.3,x.1,x.4 *>;
  coherence by Th03;
  func pi_2341(x) -> Tuple of 4,X equals <* x.2,x.3,x.4,x.1 *>;
  coherence by Th03;
  func pi_2413(x) -> Tuple of 4,X equals <* x.2,x.4,x.1,x.3 *>;
  coherence by Th03;
  func pi_2431(x) -> Tuple of 4,X equals <* x.2,x.4,x.3,x.1 *>;
  coherence by Th03;
  func pi_3124(x) -> Tuple of 4,X equals <* x.3,x.1,x.2,x.4 *>;
  coherence by Th03;
  func pi_3142(x) -> Tuple of 4,X equals <* x.3,x.1,x.4,x.2 *>;
  coherence by Th03;
  func pi_3241(x) -> Tuple of 4,X equals <* x.3,x.2,x.4,x.1 *>;
  coherence by Th03;
  func pi_3412(x) -> Tuple of 4,X equals <* x.3,x.4,x.1,x.2 *>;
  coherence by Th03;
  func pi_3421(x) -> Tuple of 4,X equals <* x.3,x.4,x.2,x.1 *>;
  coherence by Th03;
  func pi_4123(x) -> Tuple of 4,X equals <* x.4,x.1,x.2,x.3 *>;
  coherence by Th03;
  func pi_4132(x) -> Tuple of 4,X equals <* x.4,x.1,x.3,x.2 *>;
  coherence by Th03;
  func pi_4213(x) -> Tuple of 4,X equals <* x.4,x.2,x.1,x.3 *>;
  coherence by Th03;
  func pi_4312(x) -> Tuple of 4,X equals <* x.4,x.3,x.1,x.2 *>;
  coherence by Th03;
  func pi_4321(x) -> Tuple of 4,X equals <* x.4,x.3,x.2,x.1 *>;
  coherence by Th03;
end;

definition
  let X be non empty set;
  let x be Tuple of 4,X;
  func pi_id(x) -> Tuple of 4,X equals <* x.1,x.2,x.3,x.4 *>;
  coherence by Th03;
  func pi_12(x) -> Tuple of 4,X equals <* x.2,x.1,x.3,x.4 *>;
  coherence by Th03;
  involutiveness
  proof
    let x,PI be Tuple of 4,X;
    assume x = <* PI.2,PI.1,PI.3,PI.4 *>;
    then x.1 = PI.2 & x.2 = PI.1 & x.3 = PI.3 & x.4 = PI.4 & len PI=4
      by CARD_1:def 7;
    hence thesis by FINSEQ_4:76;
  end;
  func pi_13(x) -> Tuple of 4,X equals <* x.3,x.2,x.1,x.4 *>;
  coherence by Th03;
  involutiveness
  proof
    let x,PI be Tuple of 4,X;
    assume x = <* PI.3,PI.2,PI.1,PI.4 *>;
    then x.1 = PI.3 & x.2 = PI.2 & x.3 = PI.1 & x.4 = PI.4 & len PI=4
       by CARD_1:def 7;
    hence thesis by FINSEQ_4:76;
  end;
  func pi_14(x) -> Tuple of 4,X equals <* x.4,x.2,x.3,x.1 *>;
  coherence by Th03;
  involutiveness
  proof
    let x,PI be Tuple of 4,X;
    assume x = <* PI.4,PI.2,PI.3,PI.1 *>;
    then x.1 = PI.4 & x.2 = PI.2 & x.3 = PI.3 & x.4 = PI.1 & len PI=4
      by CARD_1:def 7;
    hence thesis by FINSEQ_4:76;
  end;
  func pi_23(x) -> Tuple of 4,X equals <* x.1,x.3,x.2,x.4 *>;
  coherence by Th03;
  involutiveness
  proof
    let x,PI be Tuple of 4,X;
    assume x = <* PI.1,PI.3,PI.2,PI.4 *>;
    then x.1 = PI.1 & x.2 = PI.3 & x.3 = PI.2 & x.4 = PI.4 & len PI=4
      by CARD_1:def 7;
    hence thesis by FINSEQ_4:76;
  end;
  func pi_24(x) -> Tuple of 4,X equals <* x.1,x.4,x.3,x.2 *>;
  coherence by Th03;
  involutiveness
  proof
    let x,PI be Tuple of 4,X;
    assume x = <* PI.1,PI.4,PI.3,PI.2 *>;
    then x.1 = PI.1 & x.2 = PI.4 & x.3 = PI.3 & x.4 = PI.2 & len PI=4
      by CARD_1:def 7;
    hence thesis by FINSEQ_4:76;
  end;
  func pi_34(x) -> Tuple of 4,X equals <* x.1,x.2,x.4,x.3 *>;
  coherence by Th03;
  involutiveness
  proof
    let x,PI be Tuple of 4,X;
    assume x = <* PI.1,PI.2,PI.4,PI.3 *>;
    then x.1 = PI.1 & x.2 = PI.2 & x.3 = PI.4 & x.4 = PI.3 & len PI=4
      by CARD_1:def 7;
    hence thesis by FINSEQ_4:76;
  end;
end;

registration
  let X be non empty set;
  let x be Tuple of 4,X;
  reduce pi_id(x) to x;
  reducibility
  proof
    dom x = Seg 4 by FINSEQ_2:124;
    then len x = 4 by FINSEQ_1:def 3;
    hence thesis by FINSEQ_4:76;
  end;
end;

notation
  let X be non empty set;
  let x be Tuple of 4,X;
  synonym pi_1234(x) for pi_id(x);
  synonym pi_2134(x) for pi_12(x);
  synonym pi_3214(x) for pi_13(x);
  synonym pi_4231(x) for pi_14(x);
  synonym pi_1324(x) for pi_23(x);
  synonym pi_1432(x) for pi_24(x);
  synonym pi_1243(x) for pi_34(x);
end;

theorem
  pi_12(pi_13(x)) = pi_13(pi_23(x)) &
  pi_12(pi_14(x)) = pi_14(pi_24(x)) &
  pi_12(pi_23(x)) = pi_13(pi_12(x)) &
  pi_12(pi_24(x)) = pi_14(pi_12(x)) &
  pi_12(pi_34(x)) = pi_34(pi_12(x)) &

  pi_13(pi_12(x)) = pi_23(pi_13(x)) &
  pi_13(pi_14(x)) = pi_34(pi_13(x)) &
  pi_13(pi_23(x)) = pi_12(pi_13(x)) &
  pi_13(pi_24(x)) = pi_13(pi_24(x)) &
  pi_13(pi_34(x)) = pi_14(pi_13(x)) &

  pi_23(pi_12(x)) = pi_13(pi_23(x)) &
  pi_23(pi_13(x)) = pi_12(pi_23(x)) &
  pi_23(pi_14(x)) = pi_14(pi_23(x)) &
  pi_23(pi_24(x)) = pi_34(pi_23(x)) &
  pi_23(pi_34(x)) = pi_24(pi_23(x)) &

  pi_24(pi_12(x)) = pi_14(pi_24(x)) &
  pi_24(pi_13(x)) = pi_13(pi_24(x)) &
  pi_24(pi_14(x)) = pi_12(pi_24(x)) &
  pi_24(pi_23(x)) = pi_34(pi_24(x)) &
  pi_24(pi_34(x)) = pi_23(pi_24(x)) &

  pi_34(pi_12(x)) = pi_12(pi_34(x)) &
  pi_34(pi_13(x)) = pi_14(pi_34(x)) &
  pi_34(pi_14(x)) = pi_13(pi_34(x)) &
  pi_34(pi_23(x)) = pi_24(pi_34(x)) &
  pi_34(pi_24(x)) = pi_23(pi_34(x));

theorem
  pi_1342(x) = pi_34(pi_23(x)) &
  pi_1423(x) = pi_34(pi_24(x)) &
  pi_2143(x) = pi_12(pi_34(x)) &
  pi_2314(x) = pi_23(pi_12(x)) &
  pi_2341(x) = pi_34(pi_23(pi_12(x))) &
  pi_2413(x) = pi_34(pi_24(pi_12(x))) &
  pi_2431(x) = pi_24(pi_12(x)) &
  pi_3124(x) = pi_23(pi_13(x)) &
  pi_3142(x) = pi_24(pi_34(pi_13(x))) &
  pi_3241(x) = pi_34(pi_13(x)) &
  pi_3412(x) = pi_24(pi_13(x)) &
  pi_3421(x) = pi_24(pi_23(pi_13(x))) &
  pi_4123(x) = pi_23(pi_34(pi_14(x))) &
  pi_4132(x) = pi_24(pi_14(x)) &
  pi_4213(x) = pi_34(pi_14(x)) &
  pi_4312(x) = pi_23(pi_24(pi_14(x))) &
  pi_4321(x) = pi_23(pi_14(x));

theorem
  pi_13(pi_23(pi_13(x))) = pi_12(x) &
  pi_12(pi_34(pi_23(pi_13(x)))) = pi_34(pi_23(x)) &
  pi_23(pi_24(pi_14(pi_23(pi_13(x))))) = pi_14(x);

theorem
  pi_23(pi_14(pi_34(x))) = pi_24(pi_23(pi_13(x))) &
  pi_34(pi_24(pi_12(x))) = pi_24(pi_13(pi_23(x))) &
  pi_24(pi_34(pi_13(x))) = pi_12(pi_34(pi_23(x)));

begin :: Affine-Ratio

reserve V             for RealLinearSpace,
        A,B,C,P,Q,R,S for Element of V;

theorem Th05:
  P,Q,Q are_collinear
  proof
    P in Line(P,Q) & Q in Line(P,Q) by RLTOPSP1:72;
    hence thesis;
  end;

Lm01:
  for a,b being Real st
  A = (1 - a) * P + a * Q & B = (1 - b) * P + b * Q holds
  A - B = (b - a) * (P - Q)
  proof
    let a,b be Real;
    assume that
A1: A = (1 - a) * P + a * Q and
A2: B = (1 - b) * P + b * Q;
    A - B = (1 - a) * P + (a * Q - ((1 - b) * P + b * Q))
      by A1,A2,RLVECT_1:def 3
         .= (1 - a) * P + ((a * Q - b * Q ) - (1 - b) * P)
      by RLVECT_1:27
         .= (1 - a) * P + ((a - b) * Q - ((1 - b) * P)) by RLVECT_1:35
         .= ((1 - a) * P + ((a - b) * Q)) + -((1 - b) * P) by RLVECT_1:def 3
         .= ((1 - a) * P + ((a - b) * Q)) + ((-1) * ((1 - b) * P))
      by RLVECT_1:16
         .= ((1 - a) * P + ((a - b) * Q)) + ((-1) * (1 - b)) * P
      by RLVECT_1:def 7
         .= (1 - a) * P + ((a - b) * Q + (b - 1) * P) by RLVECT_1:def 3
         .= (1 - a) * P + (((b - 1) * P) + ((a - b) * Q)) by RLVECT_1:def 2
         .= (((1 - a) * P + (b - 1) * P)) + ((a - b) * Q) by RLVECT_1:def 3
         .= (1 - a + (b - 1)) * P + (a - b) * Q by RLVECT_1:def 6
         .= (b - a) * P + (-1) * (b - a) * Q
         .= (b - a) * P + (-1) * ((b - a) * Q) by RLVECT_1:def 7
         .= (b - a) * P - ((-1) * (a - b) * Q) by RLVECT_1:16;
    hence thesis by RLVECT_1:34;
  end;

definition
  let V be RealLinearSpace;
  let A,B,C be Element of V;
  assume that
A1: A <> C and
A2: A,B,C are_collinear;
  func affine-ratio(A,B,C) -> Real means  :Def02:
  (B - A) = it * (C - A);
  existence
  proof
    consider L be line of V such that
A3: A in L and
A4: B in L and
A5: C in L by A2;
    consider P,Q be Element of V such that
A6: L = Line(P,Q) by RLTOPSP1:def 15;
A7: Line(P,Q) = the set of all (((1 - r) * P) + (r * Q)) where r is Real
      by RLTOPSP1:def 14;
    consider a be Real such that
A8: A = (1 - a) * P + a * Q by A3,A6,A7;
    consider b be Real such that
A9: B = (1 - b) * P + b * Q by A4,A6,A7;
    consider c be Real such that
A10: C = (1 - c) * P + c * Q by A5,A6,A7;
A11: a - c <> 0 by A8,A10,A1;
    set k = (a - b) / (a - c);
    B - A = (a - b) * (P - Q) by A8,A9,Lm01
         .= ((a - b) / (a - c) * (a - c)) * (P - Q) by A11,XCMPLX_1:87
         .= ((a - b) / (a - c)) * ((a - c) * (P - Q)) by RLVECT_1:def 7
         .= k * (C - A) by A8,A10,Lm01;
    hence thesis;
  end;
  uniqueness
  proof
    let r1,r2 be Real such that
A12: (B - A) = r1 * (C - A) and
A13: (B - A) = r2 * (C - A);
    C - A <> 0.V & r1 * (C - A) = r2 * (C - A)
      by A12,A13,A1,RLVECT_1:21;
    hence thesis by RLVECT_1:37;
  end;
end;

theorem
  A <> C & A,B,C are_collinear implies
  A - B = affine-ratio(A,B,C) * (A - C)
  proof
    assume that
A1: A <> C and
A2: A,B,C are_collinear;
    A - B = - (B - A) by RLVECT_1:33
         .= - affine-ratio(A,B,C) * (C - A) by Def02,A1,A2
         .= (- 1) * (affine-ratio(A,B,C) * (C - A)) by RLVECT_1:16
         .= ((- 1) * affine-ratio(A,B,C)) * (C - A) by RLVECT_1:def 7
         .= affine-ratio(A,B,C) * ((- 1) * (C - A)) by RLVECT_1:def 7
         .= affine-ratio(A,B,C) * (-(C - A)) by RLVECT_1:16
         .= affine-ratio(A,B,C) * (A - C) by RLVECT_1:33;
    hence thesis;
  end;

theorem Th06:
  A <> C & A,B,C are_collinear implies
  (affine-ratio(A,B,C) = 0 iff A = B)
  proof
    assume that
A1: A <> C and
A2: A,B,C are_collinear;
    hereby
      assume affine-ratio(A,B,C) = 0;
      then (B - A) = 0 * (C - A) by A1,A2,Def02
                  .= 0.V by RLVECT_1:10;
      hence A = B by RLVECT_1:21;
    end;
    assume A = B;
    then B - A = 0.V by RLVECT_1:5
              .= 0 * (C - A) by RLVECT_1:10;
    hence affine-ratio(A,B,C) = 0 by A1,A2,Def02;
  end;

theorem Th07:
  A <> C & A,B,C are_collinear implies
  (affine-ratio(A,B,C) = 1 iff B = C)
  proof
    assume that
A1: A <> C and
A2: A,B,C are_collinear;
    hereby
      assume affine-ratio(A,B,C) = 1;
      then (B - A) = 1 * (C - A) by A1,A2,Def02
                  .= C - A by RLVECT_1:def 8;
      hence B = C by RLVECT_1:8;
    end;
    assume B = C;
    then B - A = 1 * (C - A) by RLVECT_1:def 8;
    hence affine-ratio(A,B,C) = 1 by A1,A2,Def02;
  end;

theorem Th08:
  for a,b being Real st P <> Q & a * (P - Q) = b * (P - Q) holds a = b
  proof
    let a,b be Real;
    assume that
A1: P <> Q and
A2: a * (P - Q) = b * (P - Q);
    P - Q <> 0.V by A1, RLVECT_1:21;
    hence thesis by A2,RLVECT_1:37;
  end;

theorem Th09:
  P,Q,R are_collinear & P <> R & P <> Q implies
  affine-ratio(P,R,Q) = 1 / affine-ratio(P,Q,R)
  proof
    assume that
A1: P,Q,R are_collinear and
A2: P <> R and
A3: P <> Q;
    set r = affine-ratio(P,Q,R),
        s = affine-ratio(P,R,Q);
    P,R,Q are_collinear by A1;
    then R - P = s * (Q - P) by A3,Def02
              .= s * (r * (R - P)) by A1,A2,Def02
              .= (s * r) * (R - P) by RLVECT_1:def 7;
    then 1 * (R - P) = (s * r) * (R - P) by RLVECT_1:def 8;
    hence thesis by A2,Th08,XCMPLX_1:73;
  end;

theorem Th10:
  P,Q,R are_collinear & P <> R & Q <> R & P <> Q implies
  affine-ratio(Q,P,R) =
    affine-ratio(P,Q,R) / (affine-ratio(P,Q,R) - 1)
  proof
    assume that
A1: P,Q,R are_collinear and
A2: P <> R and
A3: Q <> R and
A4: P <> Q;
    set r = affine-ratio(P,Q,R),
        s = affine-ratio(Q,P,R);
A5: Q - P = r * (R - P) by A1,A2,Def02;
    Q,P,R are_collinear by A1;
    then P - Q = s * (R - Q) by A3,Def02;
    then r * (R - P) = -s * (R + 0.V - Q) by A5,RLVECT_1:33
                    .= -s * (R + (-P + P) - Q) by RLVECT_1:5
                    .= -s * (R - P + P - Q) by RLVECT_1:def 3
                    .= -(s * ((R - P) + (P - Q))) by RLVECT_1:def 3
                    .= (-1) * (s * ((R - P) + (P - Q))) by RLVECT_1:16
                    .= ((-1) * s) * ((R - P) + (P - Q)) by RLVECT_1:def 7
                    .= (-s) * (R - P) + (-s) * (P - Q) by RLVECT_1:def 5;
    then
A6: r * (R - P) + s * (R - P)
       = (-s) * (R - P) + ((-s) * (P - Q) + s * (R - P)) by RLVECT_1:def 3
       .= (-s) * (R - P) + (s * (R - P) + (-s) * (P - Q)) by RLVECT_1:def 2
       .= ((-s) * (R - P) + s * (R - P)) + (-s) * (P - Q) by RLVECT_1:def 3
       .= (-s + s) * (R - P) + (-s) * (P - Q) by RLVECT_1:def 6
       .= 0.V + (-s) * (P - Q) by RLVECT_1:10
       .= (-s) * (P - Q);
    Q,P,R are_collinear by A1; then
A7: s <> 0 by A3,A4,Th06;
    then reconsider s9 = 1 / s as non zero Real;
A8: s9 * s = 1 by A7,XCMPLX_1:106;
A9: r - 1 <> 0 by A1,A2,A3,Th07;
    (r + s) * (R - P) = (-s) * (P - Q) by A6,RLVECT_1:def 6
                     .= s * (-(P - Q)) by RLVECT_1:24
                     .= s * (Q - P) by RLVECT_1:33;
    then (s9 * (r + s)) * (R - P) = s9 * (s * (Q - P)) by RLVECT_1:def 7
                                 .= (s9 * s) * (Q - P) by RLVECT_1:def 7
                                 .= 1 * (Q - P) by A7,XCMPLX_1:106
                                 .= Q - P by RLVECT_1:def 8;
    then s * r  = s * ((1/s) * (r + s)) by A1,Def02,A2;
    then r * s  = (s * (1/s)) * (r + s);
    then r * s = r + s by A8;
    then s * (r - 1) / (r - 1) = r / (r - 1);
    then s * ((r - 1) / (r - 1)) = r / (r - 1);
    then s * 1 = r / (r - 1) by A9,XCMPLX_1:60;
    hence thesis;
  end;

theorem Th11:
  P,Q,R are_collinear & P <> R implies
  affine-ratio(R,Q,P) = 1 - affine-ratio(P,Q,R)
  proof
    assume that
A1: P,Q,R are_collinear and
A2: P <> R;
    set r = affine-ratio(P,Q,R),
        s = affine-ratio(R,Q,P);
A3: Q - P = r * (R - P) by A1,A2,Def02;
A4: Q - P = Q + 0.V - P
         .= Q + (-R + R) - P by RLVECT_1:5
         .= Q + (-R) + R - P by RLVECT_1:def 3
         .= (Q - R) + (R - P) by RLVECT_1:def 3
         .= (Q - R) - (P - R) by RLVECT_1:33;
    R,Q,P are_collinear by A1;
    then r * (R - P) - s * (P - R) = s * (P - R) - (P - R) - s * (P - R)
                                    by A3,A4,A2,Def02
                                  .= s * (P - R) + (R - P) - s * (P - R)
                                    by RLVECT_1:33
                                  .= s * (P - R) + ((R - P) - s * (P - R))
                                    by RLVECT_1:def 3
                                  .= s * (P - R) + (- s * (P - R) + (R - P))
                                    by RLVECT_1:def 2
                                  .= s * (P - R) - s * (P - R) + (R - P)
                                    by RLVECT_1:def 3
                                  .= 0.V + (R - P) by RLVECT_1:5
                                  .= R - P;
    then R - P = r * (R - P) - s * (-(R - P)) by RLVECT_1:33
              .= r * (R - P) - (s * ((-1) * (R - P))) by RLVECT_1:16
              .= r * (R - P) - (s * (-1)) * (R - P) by RLVECT_1:def 7
              .= r * (R - P) - (-s) * (R - P);
    then (R - P) + (-s) * (R - P)
      = r * (R - P) + (- (-s) * (R - P) + (-s) * (R - P)) by RLVECT_1:def 3
     .= r * (R - P) + 0.V by RLVECT_1:5
     .= r * (R - P);
    then 1 * (R - P) +  (-s) * (R - P) = r * (R - P) by RLVECT_1:def 8;
    then (1 - s) * (R - P) = r * (R - P) by RLVECT_1:def 6;
    then 1 - s = r by A2,Th08;
    hence thesis;
  end;

theorem Th12:
  P,Q,R are_collinear & P <> R & P <> Q implies
  affine-ratio(Q,R,P) =
  (affine-ratio(P,Q,R) - 1) / affine-ratio(P,Q,R)
  proof
    assume that
A1: P,Q,R are_collinear and
A2: P <> R and
A3: P <> Q;
    set r = affine-ratio(P,Q,R),
        s = affine-ratio(Q,R,P);
A4: r <> 0 by A1,A2,A3,Th06;
A5: Q - P = r * (R + 0.V - P) by A1,A2,Def02
         .= r * (R + (-Q + Q) - P) by RLVECT_1:5
         .= r * ((R + -Q) + Q - P) by RLVECT_1:def 3
         .= r * (R - Q + (Q - P)) by RLVECT_1:def 3
         .= r * (R - Q) + r * (Q - P) by RLVECT_1:def 5;
    Q,R,P are_collinear by A1;
    then Q - P = r * (s * (P - Q)) + r * (Q - P) by A5,A3,Def02
              .= (r * s) * (P - Q) + r * (Q - P) by RLVECT_1:def 7
              .= (r * s) * (-(Q - P)) + r * (Q - P) by RLVECT_1:33
              .= (r * s) * ((-1) * (Q - P)) + r * (Q - P) by RLVECT_1:16
              .= (r * s * (-1)) * (Q - P) + r * (Q - P) by RLVECT_1:def 7
              .= (-r * s + r) * (Q - P) by RLVECT_1:def 6;
    then 1 * (Q - P) = (r - r * s) * (Q - P) by RLVECT_1:def 8;
    then 1 = r - r * s by A3,Th08;
    hence thesis by A4,XCMPLX_1:89;
  end;

theorem Th13:
  P,Q,R are_collinear & P <> R & Q <> R implies
  affine-ratio(R,P,Q) = 1 / (1 - affine-ratio(P,Q,R))
  proof
    assume that
A1: P,Q,R are_collinear and
A2: P <> R and
A3: Q <> R;
    set r = affine-ratio(P,Q,R),
        s = affine-ratio(R,P,Q);
A4: 1 - r <> 0 by A1,A2,A3,Th07;
A5: r * (R - P) = Q + 0.V - P by A1,A2,Def02
                .= Q + (-R + R) - P by RLVECT_1:5
                .= Q - R + R - P by RLVECT_1:def 3
                .= Q - R + --(R - P) by RLVECT_1:def 3
                .= Q - R - (P - R) by RLVECT_1:33;
A6: R,P,Q are_collinear by A1;
    then - (s * (Q - R)) = - (P - R) by A3,Def02
                        .= R - P by RLVECT_1:33;
    then
A7: R - P = (-1) * (s * (Q - R)) by RLVECT_1:16
          .= ((-1) * s) * (Q - R) by RLVECT_1:def 7
          .= (-s) * (Q - R);
    r * (R - P) = (Q - R) + -(s * (Q - R)) by A5,A6,A3,Def02
               .= (Q - R) + (-1) * (s * (Q - R)) by RLVECT_1:16
               .= (Q - R) + ((-1) * s) * (Q - R) by RLVECT_1:def 7
               .= 1 * (Q - R) + (-s) * (Q - R) by RLVECT_1:def 8
               .= (1 - s) * (Q - R) by RLVECT_1:def 6;
    then (r * (-s)) * (Q - R) = (1 - s) * (Q - R) by A7,RLVECT_1:def 7;
    then - r * s = (1 - s) by Th08,A3;
    then s * (1 - r) = 1;
    hence thesis by A4,XCMPLX_1:89;
  end;

theorem
  for r being Real st P,Q,R are_collinear &
  P <> R & Q <> R & P <> Q &
  r = affine-ratio(P,Q,R) holds
  affine-ratio(P,R,Q) = 1 / r &
  affine-ratio(Q,P,R) = r / (r - 1) &
  affine-ratio(Q,R,P) = (r - 1) / r &
  affine-ratio(R,P,Q) = 1 / (1 - r) &
  affine-ratio(R,Q,P) = 1 - r by Th09,Th10,Th11,Th12,Th13;

theorem
  for a being non zero Real st P,Q,R are_collinear & P <> R holds
  affine-ratio(P,Q,R) = affine-ratio(a * P,a * Q,a * R)
  proof
    let a be non zero Real;
    assume
A1: P,Q,R are_collinear & P <> R;
    reconsider aP = a * P, aQ = a * Q, aR = a * R as Element of V;
    now
      thus aP <> aR by A1,RLVECT_1:36;
      Q in Line(P,R) by A1,RLTOPSP1:80;
      then Q in the set of all (1 - l) * P + l * R where l is Real
        by RLTOPSP1:def 14;
      then consider l be Real such that
A2:   Q = (1 - l) * P + l * R;
      reconsider aL = Line(aP,aR) as line of V;
      H1: aP in aL & aR in aL by RLTOPSP1:72;
      aQ = a * ((1 - l) * P) + a * (l * R) by A2,RLVECT_1:def 5
          .= (a * (1 - l)) * P + a * (l * R) by RLVECT_1:def 7
          .= (a * (1 - l)) * P + (a * l) * R by RLVECT_1:def 7
          .= (1 - l) * (a * P) + (a * l) * R by RLVECT_1:def 7
          .= (1 - l) * aP + l * aR by RLVECT_1:def 7;
      then aQ in the set of all (1 - l) * aP + l * aR where l is Real;
      then aQ in aL by RLTOPSP1:def 14;
      hence aP,aQ,aR are_collinear by H1;
    end;
    then aQ - aP = affine-ratio(aP,aQ,aR) * (aR - aP) by Def02;
    then a * (Q - P) = affine-ratio(aP,aQ,aR) * (aR - aP) by RLVECT_1:34;
    then a * (Q - P) = affine-ratio(aP,aQ,aR) * (a * (R - P))
      by RLVECT_1:34;
    then a * (Q - P) = (affine-ratio(aP,aQ,aR) * a) * (R - P)
      by RLVECT_1:def 7;
    then a * (Q - P) = a * (affine-ratio(aP,aQ,aR) * (R - P))
      by RLVECT_1:def 7;
    then Q - P = affine-ratio(aP,aQ,aR) * (R - P) by RLVECT_1:36;
    hence thesis by A1,Def02;
  end;

theorem
  for x,y being Element of REAL 1
  for p,q being Tuple of 1,REAL st
  p = x & q = y holds x + y = p + q;

theorem
  for x,y being Element of TOP-REAL 1
  for p,q being Tuple of 1,REAL st
  p = x & q = y holds x + y = p + q;

theorem
  for x,y being Element of TOP-REAL 1
  for p,q being Tuple of 1,REAL st
  p = x & q = y holds x - y = p - q;

theorem
  for x being Element of TOP-REAL 1
  for p being Tuple of 1,REAL st
  p = x holds - x = - p;

theorem Th14:
  for T being RealLinearSpace
  for x,y being Element of T
  for p,q being Tuple of 1,REAL st
  T = TOP-REAL 1 & p = x & q = y
  holds x + y = p + q
  proof
    let T be RealLinearSpace;
    let x,y be Element of T;
    let p,q be Tuple of 1,REAL;
    assume that
A1: T = TOP-REAL 1 and
A2: p = x and
A3: q = y;
A4: p in Funcs(Seg 1,REAL) & q is Element of Funcs(Seg 1,REAL) by SRINGS_5:11;
    (the addF of the RLSStruct of TOP-REAL 1).(p,q)
      = (the addF of RealVectSpace Seg 1).(p,q) by EUCLID:def 8
     .= p + q by A4,FUNCSDOM:def 1;
    hence thesis by A1,A2,A3,ALGSTR_0:def 1;
  end;

theorem Th15:
  for p being Tuple of 1,REAL holds - p is Tuple of 1,REAL
  proof
    let p be Tuple of 1,REAL;
    consider d be Element of REAL such that
A1: p = <* d *> by FINSEQ_2:97;
    -p = <* -d *> by A1,RVSUM_1:20;
    hence thesis;
  end;

theorem Th16:
  for T being RealLinearSpace
  for x being Element of T
  for p being Tuple of 1,REAL st
  T = TOP-REAL 1 & p = x
  holds - p = - x
  proof
    let T be RealLinearSpace;
    let x be Element of T;
    let p be Tuple of 1,REAL;
    assume that
A1: T = TOP-REAL 1 and
A2: p = x;
    consider d be Element of REAL such that
A3: p = <* d *> by FINSEQ_2:97;
    reconsider x9 = <* -d *> as Tuple of 1,REAL;
    reconsider n = 1 as Nat;
    REAL is non empty & REAL c= REAL;
    then reconsider p9 = p as Element of 1-tuples_on REAL
      by FINSEQ_2:109;
A4: the RLSStruct of TOP-REAL 1 = RealVectSpace Seg 1 & 0.REAL 1 = <* 0 *>
      by EUCLID:def 8,FINSEQ_2:59;
A5: p + (-p) = p9 + (-p9)
            .= 0.TOP-REAL 1 by A4,RVSUM_1:22;
    the TopStruct of TOP-REAL 1 = TopSpaceMetr Euclid 1 by EUCLID:def 8;
    then reconsider x99 = x9 as Element of T by A1;
A6: -p = <* -d *> by A3,RVSUM_1:20;
    x + x99 = 0.T by A1,A2,A6,Th14,A5;
    then - x = x99 by RLVECT_1:def 10;
    hence thesis by A3,RVSUM_1:20;
  end;

theorem Th17:
  for T being RealLinearSpace
  for x being Element of T
  for p being Element of TOP-REAL 1 st
  T = TOP-REAL 1 & p = x
  holds - p = - x
  proof
    let T be RealLinearSpace;
    let x be Element of T;
    let p be Element of TOP-REAL 1;
    assume that
A1: T = TOP-REAL 1 and
A2: p = x;
    p is Element of REAL 1 by EUCLID:22;
    then reconsider p9 = p as Tuple of 1,REAL;
    - p9 = - x by A1,A2,Th16;
    hence thesis;
  end;

theorem Th18:
  for T being RealLinearSpace
  for x,y being Element of T
  for p,q being Tuple of 1,REAL st
  T = TOP-REAL 1 & p = x & q = y
  holds x - y = p - q
  proof
    let T be RealLinearSpace;
    let x,y be Element of T;
    let p,q be Tuple of 1,REAL;
    assume that
A1: T = TOP-REAL 1 and
A2: p = x and
A3: q = y;
    set p9 = p, q9 = q;
    reconsider y9 = -y as Element of T;
    reconsider qm9 = -q as Tuple of 1,REAL by Th15;
A4: p9 in Funcs(Seg 1,REAL) &
      qm9 is Element of Funcs(Seg 1,REAL) by SRINGS_5:11;
A5:    (the addF of the RLSStruct of TOP-REAL 1).(p9,qm9)
      = (the addF of RealVectSpace Seg 1).(p9,qm9) by EUCLID:def 8
     .= p9 + qm9 by A4,FUNCSDOM:def 1;
    x - y = (the addF of (TOP-REAL 1)).(x,-y) by A1,ALGSTR_0:def 1
         .= p - q by A5,A1,A2,A3,Th16;
    hence thesis;
  end;

theorem Th19:
  for T being RealLinearSpace
  for x,y being Element of T
  for p,q being Element of TOP-REAL 1 st
  T = TOP-REAL 1 & p = x & q = y
  holds x + y = p + q
  proof
    let T be RealLinearSpace;
    let x,y be Element of T;
    let p,q be Element of TOP-REAL 1;
    assume that
A1: T = TOP-REAL 1 and
A2: p = x and
A3: q = y;
    reconsider p9 = p,q9 = q as Element of REAL 1 by EUCLID:22;
    x + y = p9 + q9 by A1,A2,A3,Th14;
    hence thesis;
  end;

theorem Th20:
  for D being set
  for d being Element of D holds Seg 1 --> d = <* d *>
  proof
    let D be set;
    let d be Element of D;
    Seg 1 --> d = 1 |-> d
               .= <* d *> by FINSEQ_2:59;
    hence thesis;
  end;

theorem Th21:
  for a,r being Real holds
  multreal.:( Seg 1 --> a, <* r *> ) = <* a * r *>
  proof
    let a,r be Real;
    reconsider r1 = a, r2 = r as Element of REAL by XREAL_0:def 1;
    a is Element of REAL by XREAL_0:def 1;
    then multreal.:( Seg 1 --> a, <* r *> ) = multreal.:(<* a *>, <* r *>)
                                             by Th20
                                           .= <* multreal.(r1,r2) *>
                                             by FINSEQ_2:74
                                           .= <* r1 * r2 *> by BINOP_2:def 11;
    hence thesis;
  end;

theorem Th22:
  for a being Real for p being Tuple of 1,REAL holds
  multreal.:(dom p --> a,p) = a * p
  proof
    let a be Real;
    let p be Tuple of 1,REAL;
    consider d be Element of REAL such that
A1: p = <* d *> by FINSEQ_2:97;
A2: a * p = <* a * d *> by A1,RVSUM_1:47;
    dom p = Seg 1 by A1,FINSEQ_1:def 8;
    hence thesis by A1,A2,Th21;
  end;

theorem
  for a being Real for p being Tuple of 1,REAL holds
  multreal.:(dom p --> a,p) = a * p by Th22;

theorem Th23:
  for T being RealLinearSpace for x,y being Element of T
  for a being Real for p,q being Tuple of 1,REAL st
  T = TOP-REAL 1 & p = x & q = y & x = a * y holds p = a * q
  proof
    let T be RealLinearSpace;
    let x,y be Element of T;
    let a be Real;
    let p,q be Tuple of 1,REAL;
    assume that
A1: T = TOP-REAL 1 and
A2: p = x and
A3: q = y and
A4: x = a * y;
    set p9 = q;
A5: p9 in Funcs(Seg 1,REAL) by SRINGS_5:11;
    (the Mult of the RLSStruct of TOP-REAL 1).(a,p9)
      = (the Mult of RealVectSpace Seg 1).(a,p9) by EUCLID:def 8
     .= multreal[;](a,p9) by A5,FUNCSDOM:def 3
     .= multreal.:(dom p9 --> a,p9) by FUNCOP_1:31
     .= a * p9 by Th22;
    hence thesis by A1,A2,A3,A4;
  end;

theorem
  for T being RealLinearSpace for x,y being Element of T
  for a being Real for p,q being Element of TOP-REAL 1 st
  T = TOP-REAL 1 & p = x & q = y holds x = a * y implies p = a * q
  proof
    let T be RealLinearSpace;
    let x,y be Element of T;
    let a be Real;
    let p,q be Element of TOP-REAL 1;
    assume that
A1: T = TOP-REAL 1 and
A2: p = x and
A3: q = y and
A4: x = a * y;
    p is Element of REAL 1 & q is Element of REAL 1 by EUCLID:22;
    then reconsider p9 = p,q9 = q as Tuple of 1,REAL;
    p9 = a * q9 by A1,A2,A3,A4,Th23;
    hence thesis;
  end;

theorem
  for T being RealLinearSpace for x,y being Element of T
  for p,q being Element of TOP-REAL 1 st T = TOP-REAL 1 & p = x & q = y
  holds x - y = p - q
  proof
    let T be RealLinearSpace;
    let x,y be Element of T;
    let p,q be Element of TOP-REAL 1;
    assume that
A1: T = TOP-REAL 1 and
A2: p = x and
A3: q = y;
    reconsider y9 = -y as Element of T;
    reconsider q as Element of REAL 1 by EUCLID:22;
    reconsider q9 = -q as Element of TOP-REAL 1 by EUCLID:22;
    - q = - y by A1,A3,Th17;
    hence thesis by A1,A2,Th19;
  end;

theorem Th24:
  for p,q being Tuple of 1,REAL for r being Real st
  p = r * q & p <> <* 0 *> holds ex a,b being Real st
  p = <* a *> & q =<* b *> & r = a / b
  proof
    let p,q be Tuple of 1,REAL;
    let r be Real;
    assume that
A1: p = r * q and
A2: p <> <* 0 *>;
    consider r1 be Element of REAL such that
A3: p = <* r1 *> by FINSEQ_2:97;
    consider r2 be Element of REAL such that
A4: q = <* r2 *> by FINSEQ_2:97;
    reconsider r1,r2 as Real;
    take r1,r2;
A5: <* r1 *> = <* r * r2 *> by A1,A3,A4,RVSUM_1:47;
    then
A6: r1 = r * r2 by FINSEQ_1:76;
    per cases;
    suppose
      r2 = 0;
      hence thesis by A5,A2,A3;
    end;
    suppose
A7:   r2 <> 0;
      r = r * 1
       .= r * (r2 / r2) by A7,XCMPLX_1:60
       .= r1 / r2 by A6;
      hence thesis by A3,A4;
    end;
  end;

theorem Th25:
  for x,y,z being Element of TOP-REAL 1 holds x,y,z are_collinear
  proof
    let x,y,z be Element of TOP-REAL 1;
    per cases;
    suppose
A1:   z <> y;
      reconsider L = Line(y,z) as line of TOP-REAL 1;
      H1: y in L & z in L by RLTOPSP1:72;
      reconsider x1 = x,y1 = y,z1 = z as Element of REAL 1 by EUCLID:22;
A2:   x1 in REAL 1 & y1 in REAL 1 & z1 in REAL 1;
      consider xr be Element of REAL such that
A3:   x = <* xr *> by A2,FINSEQ_2:97;
      consider yr be Element of REAL such that
A4:   y = <* yr *> by A2,FINSEQ_2:97;
      consider zr be Element of REAL such that
A5:   z = <* zr *> by A2,FINSEQ_2:97;
A6:   zr - yr <> 0 by A4,A5,A1;
      reconsider r = (xr - yr) / (zr - yr) as Real;
A7:   (1 - r) * yr + r * zr = yr + (xr -yr) / (zr - yr) * (zr - yr)
                           .= yr + (xr - yr) by A6,XCMPLX_1:87
                           .= xr;
      (1 - r) * y1 + r * z1 = <* (1 - r) * yr *> + r * <* zr *>
                                  by A4,A5,RVSUM_1:47
                           .= <* (1 - r) * yr *> + <* r * zr *> by RVSUM_1:47
                           .= x by RVSUM_1:13,A3,A7;
      then x in the set of all (1 - r) * y1 + r * z1 where r is Real;
      then x in Line(y1,z1) by EUCLID_4:def 1;
      then x in L by EUCLID12:4;
      hence thesis by H1;
    end;
    suppose z = y;
      then x in Line(x,y) & y in Line(x,y) & z in Line(x,y) by RLTOPSP1:72;
      hence thesis;
    end;
  end;

theorem
  for T being RealLinearSpace st T = TOP-REAL 1
  for x,y,z being Element of T holds x,y,z are_collinear by Th25;

theorem Th26:
  for T being RealLinearSpace st T = TOP-REAL 1
  for x,y,z being Element of T st z <> x & y <> x holds
  ex a,b,c being Real st x = <* a *> & y = <* b *> & z = <* c *> &
  affine-ratio(x,y,z) = (b - a) / (c - a)
  proof
    let T be RealLinearSpace;
    assume
A1: T = TOP-REAL 1;
    let x,y,z be Element of T;
    assume that
A2: z <> x and
A3: y <> x;
    reconsider p9 = x,q9 = y,r9 = z as Element of REAL 1 by A1,EUCLID:22;
    reconsider p = p9, q = q9,r = r9 as Tuple of 1,REAL;
    set ma = affine-ratio(x,y,z);
    reconsider yx = y - x, zx = z - x as Element of T;
    q9 - p9 is Element of 1-tuples_on REAL &
    r9 - p9 is Element of 1-tuples_on REAL;
    then reconsider qp = q - p, rp = r - p as Tuple of 1,REAL;
A4: qp = yx & rp = zx by A1,Th18;
    consider r1 be Element of REAL such that
A5: q = <* r1 *> by FINSEQ_2:97;
    consider r2 be Element of REAL such that
A6: p = <* r2 *> by FINSEQ_2:97;
    consider r3 be Element of REAL such that
A7: r = <* r3 *> by FINSEQ_2:97;
A8: qp =<* r1 - r2 *> & rp = <* r3 - r2 *> by A5,A6,A7,RVSUM_1:29;
    now
      x,y,z are_collinear by A1,Th25;
      then (y - x) = ma * (z - x) by A2,Def02;
      hence qp = ma * rp by A4,A1,Th23;
      thus qp <> <* 0 *>
      proof
        assume qp = <* 0 *>;
        then r1 - r2 = 0 by A8,FINSEQ_1:76;
        hence contradiction by A5,A6,A3;
      end;
    end;
    then consider a,b be Real such that
A9: qp = <* a *> and
A10: rp =<* b *> and
A11: ma = a / b by Th24;
    reconsider s1 = r1 - r2, s2 = r3 - r2 as Real;
A12: a = s1 & b = s2 by A9,A10,A8,FINSEQ_1:76;
    reconsider r2,r1,r3 as Real;
    take r2,r1,r3;
    thus thesis by A11,A12,A5,A6,A7;
  end;

theorem Th27:
  for x being Element of TOP-REAL 1 for a,r being Real st
  x = <* a *> holds r * x = <* (r * a) *>
  proof
    let x be Element of TOP-REAL 1;
    let a,r be Real;
    assume x = <* a *>; then
A1: x.1 = a by FINSEQ_1:def 8;
    reconsider x9 = x as Element of REAL 1 by EUCLID:22;
A2: (r * x9).1 = r * x.1 by RVSUM_1:44;
    r * x9 in REAL 1;
    then r * x is Element of TOP-REAL 1 by EUCLID:22;
    then consider b be Real such that
A3: r * x = <* b *> by JORDAN2B:20;
    thus thesis by A1,A3,A2,FINSEQ_1:def 8;
  end;

theorem
  for x,y being Element of TOP-REAL 1 for a,b,r being Real st
  x = <* a *> & y = <* b *> holds x = r * y iff a = r * b
  proof
    let x,y be Element of TOP-REAL 1;
    let a,b,r be Real;
    assume that
A1: x = <* a *> and
A2: y = <* b *>;
    reconsider rb = r * b as Real;
    hereby
      assume x = r * y;
      then x = <* rb *> by A2,Th27;
      hence a = r * b by A1,FINSEQ_1:76;
    end;
    assume a = r * b;
    hence x = r * y by A2,A1,Th27;
  end;

theorem
  for x,y being Element of TOP-REAL 1 for a,b being Real st
  x = <* a *> & y = <* b *> holds x - y = <* a - b *> by RVSUM_1:29;

theorem
  for V being RealLinearSpace for x,y being Element of F_Real
  for x9,y9 being Element of V st V = F_Real & x = x9 & y = y9 holds
  x + y = x9 + y9;

theorem
  for V being RealLinearSpace for P,Q,R being Element of V st
  P,Q,R are_collinear & P <> R & Q <> R & P <> Q holds
  affine-ratio(P,Q,R) <> 0 & affine-ratio(P,Q,R) <> 1 by Th06,Th07;

theorem Th28:
  for V being RealLinearSpace for P,Q,R being Element of V st
  P,Q,R are_collinear & P <> R & Q <> R & P <> Q holds
  ex r being non unit non zero Real st
  r = affine-ratio(P,Q,R) &
  affine-ratio(P,R,Q) = op1(r) &
  affine-ratio(Q,P,R) = op1(op2(op1(r))) &
  affine-ratio(Q,R,P) = op2(op1(r)) &
  affine-ratio(R,P,Q) = op1(op2(r)) &
  affine-ratio(R,Q,P) = op2(r)
  proof
    let V be RealLinearSpace;
    let P,Q,R be Element of V;
    assume that
A1: P,Q,R are_collinear and
A2: P <> R and
A3: Q <> R and
A4: P <> Q;
A5: affine-ratio(P,Q,R) <> 0 & affine-ratio(P,Q,R) <> 1
      by A1,A2,A3,A4,Th06,Th07;
    reconsider r = affine-ratio(P,Q,R) as Element of REAL
      by XREAL_0:def 1;
    reconsider r9 = r as non unit non zero Real by A5,Def01;
    take r9;
    affine-ratio(P,R,Q) = 1 / r &
      affine-ratio(Q,P,R) = r / (r - 1) &
      affine-ratio(Q,R,P) = (r - 1) / r &
      affine-ratio(R,P,Q) = 1 / (1 - r) &
      affine-ratio(R,Q,P) = 1 - r
      by A1,A2,A3,A4,Th09,Th10,Th11,Th12,Th13;
    hence thesis by Th01;
  end;

begin :: Cross-Ratio

theorem
  for X being non empty set
  for x being Tuple of 4,X
  for P,Q,R,S being Element of X st x = <* P,Q,R,S *> holds
  pi_1234(x) = <* P, Q, R, S *> &
  pi_1243(x) = <* P, Q, S, R *> &
  pi_1324(x) = <* P, R, Q, S *> &
  pi_1342(x) = <* P, R, S, Q *> &
  pi_1423(x) = <* P, S, Q, R *> &
  pi_1432(x) = <* P, S, R, Q *> &
  pi_2134(x) = <* Q, P, R, S *> &
  pi_2143(x) = <* Q, P, S, R *> &
  pi_2314(x) = <* Q, R, P, S *> &
  pi_2341(x) = <* Q, R, S, P *> &
  pi_2413(x) = <* Q, S, P, R *> &
  pi_2431(x) = <* Q, S, R, P *> &
  pi_3124(x) = <* R, P, Q, S *> &
  pi_3142(x) = <* R, P, S, Q *> &
  pi_3214(x) = <* R, Q, P, S *> &
  pi_3241(x) = <* R, Q, S, P *> &
  pi_3412(x) = <* R, S, P, Q *> &
  pi_3421(x) = <* R, S, Q ,P *> &
  pi_4123(x) = <* S, P, Q, R *> &
  pi_4132(x) = <* S, P, R, Q *> &
  pi_4213(x) = <* S, Q, P, R *> &
  pi_4231(x) = <* S, Q, R, P *> &
  pi_4312(x) = <* S, R, P, Q *> &
  pi_4321(x) = <* S, R, Q, P *>;

theorem
  for X being non empty set
  for x being Tuple of 4,X holds
  pi_1324(pi_1243(x)) = pi_1423(x) & pi_2143(pi_1243(x)) = pi_2134(x) &
  pi_3412(pi_1243(x)) = pi_4312(x) & pi_4321(pi_1243(x)) = pi_3421(x) &
  pi_3412(pi_1324(x)) = pi_2413(x) & pi_2143(pi_1324(x)) = pi_3142(x) &
  pi_4321(pi_1324(x)) = pi_4231(x) & pi_3412(pi_1423(x)) = pi_2314(x) &
  pi_2143(pi_1423(x)) = pi_4132(x) & pi_4321(pi_1423(x)) = pi_3241(x) &
  pi_1243(pi_1423(x)) = pi_1432(x) & pi_4321(pi_1432(x)) = pi_2341(x) &
  pi_3412(pi_1432(x)) = pi_3214(x) & pi_2143(pi_1432(x)) = pi_4123(x) &
  pi_4321(pi_3124(x)) = pi_4213(x) & pi_3412(pi_3124(x)) = pi_2431(x) &
  pi_2143(pi_3124(x)) = pi_1342(x) & pi_4312(pi_3124(x)) = pi_4231(x) &
  pi_4321(pi_3124(x)) = pi_4213(x);

reserve x           for Tuple of 4,the carrier of V,
        P9,Q9,R9,S9 for Element of V;

definition
  let V being RealLinearSpace;
  let P,Q,R,S being Element of V;
  func cross-ratio(P,Q,R,S) -> Real equals
  affine-ratio(R,P,Q) / affine-ratio(S,P,Q);
  coherence;
end;

theorem Th31:
  P,Q,R,S are_collinear & R <> Q & S <> Q & S <> P implies
  (R = P iff cross-ratio(P,Q,R,S) = 0)
  proof
    assume that
A1: P,Q,R,S are_collinear and
A2: R <> Q and
A3: S <> Q and
A4: S <> P;
    consider L be line of V such that
A5: P in L & Q in L & R in L & S in L by A1;
A6: R,P,Q are_collinear & S,P,Q are_collinear by A5;
    hereby
      assume R = P;
      then affine-ratio(R,P,Q) = 0 by A6,A2,Th06;
      hence cross-ratio(P,Q,R,S) = 0;
    end;
    assume
A7: cross-ratio(P,Q,R,S) = 0;
    per cases;
    suppose affine-ratio(S,P,Q) = 0;
      hence R = P by A3,A4,A6,Th06;
    end;
    suppose affine-ratio(S,P,Q) <> 0;
      then affine-ratio(R,P,Q) = 0 by A7;
      hence R = P by A2,A6,Th06;
    end;
  end;

theorem
  P <> R & P <> S implies cross-ratio(P,P,R,S) = 1
  proof
    assume that
A1: P <> R and
A2: P <> S;
    R,P,P are_collinear & S,P,P are_collinear by Th05;
    then affine-ratio(R,P,P) = 1 & affine-ratio(S,P,P) = 1
      by A1,A2,Th07;
    hence thesis;
  end;

theorem Th32:
  P,Q,R,S are_collinear & R <> Q & S <> Q & R <> S &
  cross-ratio(P,Q,R,S) = 1 implies P = Q
  proof
    assume that
A1: P,Q,R,S are_collinear and
A2: R <> Q and
A3: S <> Q and
A4: R <> S and
A5: cross-ratio(P,Q,R,S) = 1;
A6: affine-ratio(R,P,Q) = affine-ratio(S,P,Q) by A5,XCMPLX_1:58;
    set r = affine-ratio(R,P,Q);
A7: R,P,Q are_collinear & S,P,Q are_collinear by A1;
    then P + 0.V - R = r * (Q + 0.V - R) by A2,Def02;
    then P + (-S + S) - R = r * (Q + 0.V - R) by RLVECT_1:5
                         .= r * (Q + (-S + S) - R) by RLVECT_1:5
                         .= r * (Q - S + S - R) by RLVECT_1:def 3;
    then P - S + S - R = r * (Q - S + S - R) by RLVECT_1:def 3
                      .= r * ((Q - S) + (S - R)) by RLVECT_1:def 3
                      .= r * (Q - S) + r * (S - R) by RLVECT_1:def 5
                      .= (P - S) + r * (S - R) by A7,A6,A3,Def02;
    then -(P - S) + ((P - S) + (S - R)) = -(P - S) + ((P - S) + r * (S - R))
      by RLVECT_1:def 3;
    then (-(P - S) + (P - S)) + (S - R) = -(P - S) + ((P - S) + r * (S - R))
                                           by RLVECT_1:def 3
                                       .= (-(P - S) + (P - S)) + r * (S - R)
                                           by RLVECT_1:def 3;
    then r * (S - R) = S - R by RLVECT_1:8
                    .= 1 * (S - R) by RLVECT_1:def 8;
    then r = 1 by A4,Th08;
    hence P = Q by A2,Th07,A7;
  end;

theorem
  P,Q,R,S are_collinear & P9,Q9,R9,S9 are_collinear & S <> P & S <> Q &
  S9 <> P9 & S9 <> Q9 implies
  (cross-ratio(P,Q,R,S) = cross-ratio(P9,Q9,R9,S9) iff
  affine-ratio(R,P,Q) * affine-ratio(S9,P9,Q9) =
  affine-ratio(R9,P9,Q9) * affine-ratio(S,P,Q))
  proof
    assume that
A1: P,Q,R,S are_collinear and
A2: P9,Q9,R9,S9 are_collinear and
A3: S <> P and
A4: S <> Q and
A5: S9 <> P9 and
A6: S9 <> Q9;
    set r  = affine-ratio(R,P,Q),     s = affine-ratio(S,P,Q),
        r9 = affine-ratio(R9,P9,Q9), s9 = affine-ratio(S9,P9,Q9);
    S,P,Q are_collinear & S9,P9,Q9 are_collinear by A1,A2;
    then s <> 0 & s9 <> 0 by A3,A4,A5,A6,Th06;
    hence thesis by XCMPLX_1:94,95;
  end;

Lm02:
  for r being Real st
  P - Q = r * (R - S) holds Q - P = r * (S - R)
  proof
    let r be Real;
    assume
A1: P - Q = r * (R - S);
    Q - P = - r * (R - S) by A1,RLVECT_1:33
         .= r * (-(R-S)) by RLVECT_1:25
         .= r * (S - R) by RLVECT_1:33;
    hence thesis;
  end;

theorem Th33:
  P,Q,R,S are_collinear & P <> S & R <> Q & S <> Q implies
  cross-ratio(P,Q,R,S) = cross-ratio(R,S,P,Q)
  proof
    assume that
A1: P,Q,R,S are_collinear and
A2: P <> S and
A3: R <> Q and
A4: S <> Q;
A5: R,P,Q are_collinear & P,R,S are_collinear & S,P,Q are_collinear &
      Q,R,S are_collinear by A1;
    set r1 = affine-ratio(R,P,Q), r2 = affine-ratio(S,P,Q),
        s1 = affine-ratio(P,R,S), s2 = affine-ratio(Q,R,S);
A6: r2 <> 0 & s2 <> 0 by A2,A3,A4,A5,Th06;
A7: Q - S <> 0.V by A4,RLVECT_1:21;
A8: P - R = r1 * (Q - R) by A5,A3,Def02;
A9: P - S = r2 * (Q - S) by A5,A4,Def02;
    R - Q = s2 * (S - Q) by A5,A4,Def02; then
A10: Q - R = s2 * (Q - S) by Lm02;
    R - P = s1 * (S - P) by A5,A2,Def02;
    then P - R = s1 * (P - S) by Lm02
              .= s1 * r2 * (Q - S) by A9,RLVECT_1:def 7;
    then r1 * s2 * (Q - S) = s1 * r2 * (Q - S) by A8,A10,RLVECT_1:def 7;
    hence thesis by A7,RLVECT_1:37,A6,XCMPLX_1:94;
  end;

theorem Th34:
  for V being RealLinearSpace for P,Q,R,S being Element of V st
  P,Q,R,S are_collinear & P <> R & P <> S & R <> Q & S <> Q holds
  cross-ratio(P,Q,R,S) = cross-ratio(Q,P,S,R)
  proof
    let V be RealLinearSpace;
    let P,Q,R,S be Element of V;
    assume that
A1: P,Q,R,S are_collinear and
A2: P <> R and
A3: P <> S and
A4: R <> Q and
A5: S <> Q;
    set r1 = affine-ratio(R,P,Q), r2 = affine-ratio(S,P,Q),
        s1 = affine-ratio(S,Q,P), s2 = affine-ratio(R,Q,P);
    per cases;
    suppose
A6:   P = Q;
      R,P,P are_collinear & S,P,P are_collinear by Th05;
      then r1 = 1 & r2 = 1 & s1 = 1 & s2 = 1 by A2,A3,A6,Th07;
      hence thesis;
    end;
    suppose
A7:   P <> Q;
      P,Q,R are_collinear by A1;
      then consider r9 be non unit non zero Real such that
A8:   r9 = affine-ratio(P,Q,R) &
      affine-ratio(P,R,Q) = op1(r9) &
      affine-ratio(Q,P,R) = op1(op2(op1(r9))) &
      affine-ratio(Q,R,P) = op2(op1(r9)) &
      affine-ratio(R,P,Q) = op1(op2(r9)) &
      affine-ratio(R,Q,P) = op2(r9) by A2,A4,A7,Th28;
      P,Q,S are_collinear & P <> S & Q <> S by A1,A3,A5;
      then ex s9 be non unit non zero Real st
        s9 = affine-ratio(P,Q,S) &
        affine-ratio(P,S,Q) = op1(s9) &
        affine-ratio(Q,P,S) = op1(op2(op1(s9))) &
        affine-ratio(Q,S,P) = op2(op1(s9)) &
        affine-ratio(S,P,Q) = op1(op2(s9)) &
        affine-ratio(S,Q,P) = op2(s9) by A7,Th28;
      hence thesis by A8;
    end;
  end;

theorem Th34bis:
  P,Q,R,S are_collinear & P <> R & P <> S & R <> Q & S <> Q implies
  cross-ratio(P,Q,R,S) = cross-ratio(S,R,Q,P)
  proof
    assume that
A1: P,Q,R,S are_collinear and
A2: P <> R and
A3: P <> S and
A4: R <> Q and
A5: S <> Q;
    R,S,P,Q are_collinear by A1;
    then cross-ratio(R,S,P,Q) = cross-ratio(S,R,Q,P) by A2,A3,A4,A5,Th34;
    hence thesis by A1,A3,A4,A5,Th33;
  end;

theorem
  cross-ratio(P,Q,S,R) = 1 / cross-ratio(P,Q,R,S) by XCMPLX_1:57;

theorem
  P,Q,R,S are_collinear & P <> R & P <> S & R <> Q & S <> Q implies
  cross-ratio(Q,P,R,S) = 1 / cross-ratio(P,Q,R,S)
  proof
    assume that
A1: P,Q,R,S are_collinear and
A2: P <> R and
A3: P <> S and
A4: R <> Q and
A5: S <> Q;
A6: cross-ratio(P,Q,S,R) = 1 / cross-ratio(P,Q,R,S) by XCMPLX_1:57;
    Q,P,R,S are_collinear & P <> R & P <> S & R <> Q & S <> Q
      by A2,A3,A4,A5,A1;
    hence thesis by A6,Th34;
  end;

theorem
  P,Q,R,S are_collinear & P <> R & P <> S & R <> Q & S <> Q implies
  cross-ratio(R,S,Q,P) = 1 / cross-ratio(P,Q,R,S)
  proof
    assume that
A1: P,Q,R,S are_collinear and
A2: P <> R and
A3: P <> S and
A4: R <> Q and
A5: S <> Q;
A6: cross-ratio(P,Q,S,R) = 1 / cross-ratio(P,Q,R,S) by XCMPLX_1:57;
    P,Q,S,R are_collinear & P <> R & P <> S & R <> Q & S <> Q
      by A1,A2,A3,A4,A5;
    hence thesis by A6,Th34bis;
  end;

theorem
  P,Q,R,S are_collinear & P <> R & P <> S & R <> Q & S <> Q implies
  cross-ratio(S,R,P,Q) = 1 / cross-ratio(P,Q,R,S)
  proof
    assume that
A1: P,Q,R,S are_collinear and
A2: P <> R and
A3: P <> S and
A4: R <> Q and
A5: S <> Q;
A6: cross-ratio(P,Q,S,R) = 1 / cross-ratio(P,Q,R,S) by XCMPLX_1:57;
    S,R,P,Q are_collinear & P <> R & P <> S & R <> Q & S <> Q
      by A1,A2,A3,A4,A5;
    hence thesis by A6,Th33;
  end;

theorem Th35:
  P,Q,R,S are_collinear & P,Q,R,S are_mutually_distinct implies
  cross-ratio(P,R,Q,S) = 1 - cross-ratio(P,Q,R,S)
  proof
    assume that
A1: P,Q,R,S are_collinear and
A2: P,Q,R,S are_mutually_distinct;
A3: P <> R & P <> S & R <> Q & S <> Q & S <> R & P <> Q  by A2,ZFMISC_1:def 6;
    set r1 = affine-ratio(R,P,Q), r2 = affine-ratio(S,P,Q),
        s1 = affine-ratio(Q,P,R), s2 = affine-ratio(S,P,R),
         r = affine-ratio(P,Q,R);
A4: P,Q,R are_collinear by A1;
    then
A5: r1 = 1 / (1 -r) & s1 = r / (r - 1) by A3,Th10,Th13;
    P,Q,R are_collinear by A1;
    then
A6: r - 1 <> 0 by A3,Th07;
A7: 1 - s1 = r1
    proof
      1 - s1 = 1 - r / (r - 1) by A4,A3,Th10
            .= (r - 1) / (r - 1) - r / (r - 1) by A6,XCMPLX_1:60
            .= (-1) / (r - 1)
            .= 1 / (-(r - 1)) by XCMPLX_1:192;
      hence thesis by A5;
    end;
    R <> Q & R,P,Q are_collinear by A1,A2,ZFMISC_1:def 6;
    then
A8: r2 * (P - R) = r2 * (r1 * (Q - R)) by Def02;
A9: S <> Q & S,P,Q are_collinear by A1,A2,ZFMISC_1:def 6;
A10: S <> R & S,P,R are_collinear by A1,A2,ZFMISC_1:def 6; then
A11: P - S = s2 * (R - S) by Def02;
    S,P,Q are_collinear by A1; then
A12: r2 <> 0 by A3,Th06;
    S,P,R are_collinear by A1; then
A13: s2 <> 0 by A3,Th06;
    P - R = P + 0.V - R
         .= P + (-S + S) - R by RLVECT_1:5
         .= ((P - S) + S) - R by RLVECT_1:def 3
         .= (P - S) + (S - R) by RLVECT_1:def 3
         .= s2 * (R - S) + (S - R) by A10,Def02
         .= s2 * (R - S) + -(R - S) by RLVECT_1:33
         .= s2 * (R - S) + (-1) * (R - S) by RLVECT_1:16
         .= (s2 - 1) * (R - S) by RLVECT_1:def 6;
    then
A14: r2 * (P - R) = (r2 * (s2 - 1)) * (R - S) by RLVECT_1:def 7;
    r1 * (Q - R) = r1 * (Q + 0.V - R)
                .= r1 * (Q + (-S + S) - R) by RLVECT_1:5
                .= r1 * (Q - S + S - R) by RLVECT_1:def 3
                .= r1 * ((Q - S) + (S - R)) by RLVECT_1:def 3
                .= r1 * (Q - S) + r1 * (S - R) by RLVECT_1:def 5
                .= r1 * (Q - S) + r1 * ( - (R - S)) by RLVECT_1:33
                .= r1 * (Q - S) + r1 * ((-1) * (R - S)) by RLVECT_1:16
                .= r1 * (Q - S) + (r1 * (-1)) * (R - S) by RLVECT_1:def 7
                .= r1 * (Q - S) + (-r1) * (R - S);
    then r2 * (r1 * (Q - R)) = r2 * (r1 * ((Q - S))) + r2 * ((-r1) * (R - S))
                              by RLVECT_1:def 5
                            .= (r2 * r1) * (Q - S) + r2 * ((-r1) * (R - S))
                              by RLVECT_1:def 7
                            .= r1 * (r2 * (Q - S)) + r2 * ((-r1) * (R - S))
                              by RLVECT_1:def 7
                            .= r1 * (s2 * (R - S)) + r2 * ((-r1) * (R - S))
                              by A9,Def02,A11
                            .= (r1 * s2) * (R - S) + r2 * ((-r1) * (R - S))
                              by RLVECT_1:def 7
                            .= (r1 * s2) * (R - S) + (r2 * (-r1)) * (R - S)
                              by RLVECT_1:def 7
                            .= (r1 * s2 + (r2 * (-r1))) * (R - S)
                              by RLVECT_1:def 6;
    then r2 * (s2 - 1) = r1 * s2 + (r2 * (-r1)) by A14,A8,A3,Th08;
    then r1 * r2 - r2 = r1 * s2 - r2 * s2;
    then (-s1) * r2 = (r1 - r2) * s2 by A7;
    then s1 * r2 = (r2 - r1) * s2;
    then s1 / s2 = (r2 - r1) / r2 by A12,A13,XCMPLX_1:94
                .= r2 / r2 - r1 / r2
                .= 1 - r1 / r2 by A12,XCMPLX_1:60;
    hence thesis;
  end;

theorem
  P,Q,R,S are_collinear & P,Q,R,S are_mutually_distinct implies
  cross-ratio(Q,S,P,R) = 1 - cross-ratio(P,Q,R,S)
  proof
    assume that
A1: P,Q,R,S are_collinear and
A2: P,Q,R,S are_mutually_distinct;
A3: P <> R & P <> S & R <> Q & S <> Q & S <> R & P <> Q  by A2,ZFMISC_1:def 6;
    P,R,Q,S are_collinear by A1;
    then cross-ratio(P,R,Q,S) = cross-ratio(Q,S,P,R) by A3,Th33;
    hence thesis by A1,A2,Th35;
  end;

theorem
  P,Q,R,S are_collinear & P,Q,R,S are_mutually_distinct implies
  cross-ratio(R,P,S,Q) = 1 - cross-ratio(P,Q,R,S)
  proof
    assume that
A1: P,Q,R,S are_collinear and
A2: P,Q,R,S are_mutually_distinct;
A3: P <> R & P <> S & R <> Q & S <> Q & S <> R & P <> Q  by A2,ZFMISC_1:def 6;
    P,R,Q,S are_collinear by A1;
    then cross-ratio(P,R,Q,S) = cross-ratio(R,P,S,Q) by A3,Th34;
    hence thesis by A1,A2,Th35;
  end;

theorem
  P,Q,R,S are_collinear & P,Q,R,S are_mutually_distinct implies
  cross-ratio(S,Q,R,P) = 1 - cross-ratio(P,Q,R,S)
  proof
    assume that
A1: P,Q,R,S are_collinear and
A2: P,Q,R,S are_mutually_distinct;
A3: P <> R & P <> S & R <> Q & S <> Q & S <> R & P <> Q  by A2,ZFMISC_1:def 6;
    P,R,Q,S are_collinear by A1;
    then cross-ratio(S,Q,R,P) = cross-ratio(P,R,Q,S)
      by A3,Th34bis;
    hence thesis by A1,A2,Th35;
  end;

definition
  let V being RealLinearSpace;
  let x being Tuple of 4,the carrier of V;
  func cross-ratio-tuple(x) -> Real means :Def03:
  ex P,Q,R,S being Element of V st P = x.1 & Q = x.2 & R = x.3 &
  S = x.4 & it = cross-ratio(P,Q,R,S);
  existence
  proof
    dom x = Seg 4 by FINSEQ_2:124;
    then 1 in dom x & 2 in dom x & 3 in dom x & 4 in dom x;
    then reconsider P = x.1, Q = x.2, R = x.3, S = x.4 as Element of V
      by FINSEQ_2:11;
    cross-ratio(P,Q,R,S) is Real;
    hence thesis;
  end;
  uniqueness;
end;

theorem Th36:
  x = <* P, Q, R, S *> implies
  cross-ratio(P,Q,R,S) = cross-ratio-tuple(x)
  proof
    assume x = <* P, Q, R, S *>;
    then x.1 = P & x.2 = Q & x.3 = R & x.4 = S;
    hence thesis by Def03;
  end;

theorem Th37:
  x = <* P, Q, R, S *> & P,Q,R,S are_collinear &
  P <> S & Q <> R & Q <> S implies
  cross-ratio-tuple(x) = cross-ratio-tuple(pi_3412(x))
  proof
    assume that
A1: x = <* P, Q, R, S *> and
A2: P,Q,R,S are_collinear and
A3: P <> S and
A4: Q <> R and
A5: Q <> S;
    consider P9,Q9,R9,S9 be Element of V such that
A7: P9 = x.1 & Q9 = x.2 & R9 = x.3 & S9 = x.4 &
      cross-ratio-tuple(x) = cross-ratio(P9,Q9,R9,S9) by Def03;
    ex P99,Q99,R99,S99 be Element of V st
    P99 = (pi_3412(x)).1 & Q99 = (pi_3412(x)).2 &
      R99 = (pi_3412(x)).3 & S99 = (pi_3412(x)).4 &
      cross-ratio-tuple(pi_3412(x)) = cross-ratio(P99,Q99,R99,S99) by Def03;
    hence thesis by A1,A7,A2,A3,A4,A5,Th33;
  end;

theorem Th38:
  x = <* P, Q, R, S *> & P,Q,R,S are_collinear & P <> R & P <> S &
  Q <> R & Q <> S implies
  cross-ratio-tuple(x) = cross-ratio-tuple(pi_2143(x)) &
  cross-ratio-tuple(x) = cross-ratio-tuple(pi_4321(x))
  proof
    assume that
A1: x = <* P, Q, R, S *> and
A2: P,Q,R,S are_collinear and
A3: P <> R and
A4: P <> S and
A5: Q <> R and
A6: Q <> S;
A8: ex P9,Q9,R9,S9 be Element of V st P9 = x.1 & Q9 = x.2 & R9 = x.3 &
    S9 = x.4 & cross-ratio-tuple(x) = cross-ratio(P9,Q9,R9,S9) by Def03;
    H1: ex P99,Q99,R99,S99 be Element of V st P99 = (pi_2143(x)).1 &
        Q99 = (pi_2143(x)).2 & R99 = (pi_2143(x)).3 & S99 = (pi_2143(x)).4 &
        cross-ratio-tuple(pi_2143(x)) = cross-ratio(P99,Q99,R99,S99) by Def03;
    ex P99,Q99,R99,S99 be Element of V st P99 = (pi_4321(x)).1 &
        Q99 = (pi_4321(x)).2 & R99 = (pi_4321(x)).3 & S99 = (pi_4321(x)).4 &
        cross-ratio-tuple(pi_4321(x)) = cross-ratio(P99,Q99,R99,S99) by Def03;
    hence thesis by H1,A6,A1,A8,A2,A3,A4,A5,Th34,Th34bis;
  end;

theorem Th39:
  cross-ratio-tuple(pi_1243(x)) = 1 / cross-ratio-tuple(x)
   proof
     consider P9,Q9,R9,S9 be Element of V such that
A1:  P9 = x.1 & Q9 = x.2 & R9 = x.3 & S9 = x.4 &
       cross-ratio-tuple(x) = cross-ratio(P9,Q9,R9,S9) by Def03;
     ex P99,Q99,R99,S99 be Element of V st
     P99 = (pi_1243(x)).1 & Q99 = (pi_1243(x)).2 &
       R99 = (pi_1243(x)).3 & S99 = (pi_1243(x)).4 &
       cross-ratio-tuple(pi_1243(x)) = cross-ratio(P99,Q99,R99,S99) by Def03;
     hence thesis by A1,XCMPLX_1:57;
   end;

theorem
  x = <*P,Q,R,S*> & P,Q,R,S are_mutually_distinct &
  P,Q,R,S are_collinear implies
  (ex r being non unit non zero Real st r = cross-ratio-tuple(x) &
  cross-ratio-tuple(pi_1243(x)) = op1(r))
  proof
    assume that
A1: x = <*P,Q,R,S*> and
A2: P,Q,R,S are_mutually_distinct and
A3: P,Q,R,S are_collinear;
A4: P <> R & P <> S & R <> Q & S <> Q & S <> R & P <> Q
      by A2,ZFMISC_1:def 6;
    consider P9,Q9,R9,S9 be Element of V such that
A5: P9 = x.1 & Q9 = x.2 & R9 = x.3 & S9 = x.4 &
      cross-ratio-tuple(x) = cross-ratio(P9,Q9,R9,S9) by Def03;
    reconsider r=cross-ratio-tuple(x) as non unit non zero Real
      by Def01,A1,A3,A5,A4,Th32,Th31;
    take r;
    thus thesis by Th39;
  end;

theorem Th40:
  x = <*P,Q,R,S*> & P,Q,R,S are_collinear & P <> R & P <> S &
  Q <> R & Q <> S implies
  cross-ratio-tuple(pi_1243(x)) = 1 / cross-ratio-tuple(x) &
  cross-ratio-tuple(pi_2134(x)) = 1 / cross-ratio-tuple(x) &
  cross-ratio-tuple(pi_3421(x)) = 1 / cross-ratio-tuple(x) &
  cross-ratio-tuple(pi_4312(x)) = 1 / cross-ratio-tuple(x)
  proof
    assume that
A1: x = <*P,Q,R,S*> and
A2: P,Q,R,S are_collinear and
A3: P <> R and
A4: P <> S and
A5: Q <> R and
A6: Q <> S;
A7: pi_1243(x) = <*P,Q,S,R*> & P,Q,S,R are_collinear by A2,A1;
    consider P9,Q9,R9,S9 be Element of V such that
A8: P9 = x.1 & Q9 = x.2 & R9 = x.3 & S9 = x.4 &
      cross-ratio-tuple(x) = cross-ratio(P9,Q9,R9,S9) by Def03;
    consider P99,Q99,R99,S99 be Element of V such that
A9: P99 = (pi_1243(x)).1 & Q99 = (pi_1243(x)).2 &
     R99 = (pi_1243(x)).3 & S99 = (pi_1243(x)).4 &
     cross-ratio-tuple(pi_1243(x)) = cross-ratio(P99,Q99,R99,S99) by Def03;
    now
      thus cross-ratio-tuple(pi_2134(x))
        = cross-ratio-tuple(pi_2143(pi_1243(x)))
        .= cross-ratio-tuple(pi_1243(x)) by A7,A3,A4,A5,A6,Th38;
      thus cross-ratio-tuple(pi_3421(x))
        = cross-ratio-tuple(pi_4321(pi_1243(x)))
        .= cross-ratio-tuple(pi_1243(x)) by A7,A3,A4,A5,A6,Th38;
      thus cross-ratio-tuple(pi_4312(x))
        = cross-ratio-tuple(pi_3412(pi_1243(x)))
        .= cross-ratio-tuple(pi_1243(x)) by A7,A3,A5,A6,Th37;
    end;
    hence thesis by A8,A9,XCMPLX_1:57;
  end;

theorem Th41:
  x = <* P,Q,R,S *> & P,Q,R,S are_mutually_distinct &
  P,Q,R,S are_collinear implies
  cross-ratio-tuple(pi_1324(x)) = 1 - cross-ratio-tuple(x) &
  cross-ratio-tuple(pi_2413(x)) = 1 - cross-ratio-tuple(x) &
  cross-ratio-tuple(pi_3142(x)) = 1 - cross-ratio-tuple(x) &
  cross-ratio-tuple(pi_4231(x)) = 1 - cross-ratio-tuple(x)
  proof
    assume that
A1: x = <* P,Q,R,S *> and
A2: P,Q,R,S are_mutually_distinct and
A3: P,Q,R,S are_collinear;
A4: P <> R & P <> S & R <> Q & S <> Q & S <> R & P <> Q
      by A2,ZFMISC_1:def 6;
A6: pi_1324(x) = <*P,R,Q,S*> & P,R,Q,S are_collinear by A3,A1;
    now
      x.1 = P & x.2 = Q & x.3 = R & x.4 = S & (pi_1324(x)).1 = P &
       (pi_1324(x)).2 = R & (pi_1324(x)).3 = Q & (pi_1324(x)).4 = S by A1;
      hence cross-ratio-tuple(pi_1324(x)) = cross-ratio(P,R,Q,S) &
        cross-ratio-tuple(x) = cross-ratio(P,Q,R,S) by Def03;
      thus cross-ratio-tuple(pi_2413(x))
        = cross-ratio-tuple(pi_3412(pi_1324(x)))
       .= cross-ratio-tuple(pi_1324(x)) by A6,A4,Th37;
      thus cross-ratio-tuple(pi_3142(x))
        = cross-ratio-tuple(pi_2143(pi_1324(x)))
       .= cross-ratio-tuple(pi_1324(x)) by A4,Th38,A6;
      thus cross-ratio-tuple(pi_4231(x))
        = cross-ratio-tuple(pi_4321(pi_1324(x)))
       .= cross-ratio-tuple(pi_1324(x)) by A4,Th38,A6;
    end;
    hence thesis by A2,A3,Th35;
  end;

theorem
  x = <* P,Q,R,S *> & P,Q,R,S are_mutually_distinct &
  P,Q,R,S are_collinear implies
  cross-ratio-tuple(pi_3124(x)) = 1 / (1 - cross-ratio-tuple(x)) &
  cross-ratio-tuple(pi_2431(x)) = 1 / (1 - cross-ratio-tuple(x)) &
  cross-ratio-tuple(pi_1342(x)) = 1 / (1 - cross-ratio-tuple(x)) &
  cross-ratio-tuple(pi_4213(x)) = 1 / (1 - cross-ratio-tuple(x))
  proof
    assume that
A1: x = <* P,Q,R,S *> and
A2: P,Q,R,S are_mutually_distinct and
A3: P,Q,R,S are_collinear;
A4: P <> R & P <> S & R <> Q & S <> Q & S <> R & P <> Q
      by A2,ZFMISC_1:def 6;
A7: cross-ratio-tuple(pi_1243(pi_3142(x))) = 1 / cross-ratio-tuple(pi_3142(x))
      by Th39;
    hence cross-ratio-tuple(pi_3124(x)) = 1 / (1 - cross-ratio-tuple(x))
      by A2,A3,A1,Th41;
A8: pi_3124(x) = <* R,P,Q,S *> & R,P,Q,S are_collinear by A3,A1;
    now
      thus cross-ratio-tuple(pi_3412(pi_3124(x)))
        = cross-ratio-tuple(pi_3124(x)) by A8,Th37,A4
       .= 1 / (1 - cross-ratio-tuple(x)) by A3,A7,A1,A2,Th41;
      thus cross-ratio-tuple(pi_2143(pi_3124(x)))
        = cross-ratio-tuple(pi_3124(x)) by A8,A4,Th38
       .= 1 / (1 - cross-ratio-tuple(x))
         by A3,A7,A1,A2,Th41;
      thus cross-ratio-tuple(pi_4321(pi_3124(x)))
        = cross-ratio-tuple(pi_3124(x)) by A4,A8,Th38
       .= 1 / (1 - cross-ratio-tuple(x)) by A3,A7,A1,A2,Th41;
    end;
    hence thesis;
  end;

theorem Th42:
  x = <* P,Q,R,S *> & P,Q,R,S are_mutually_distinct &
  P,Q,R,S are_collinear implies
  cross-ratio-tuple(pi_1423(x))
    = (cross-ratio-tuple(x) - 1) / cross-ratio-tuple(x) &
  cross-ratio-tuple(pi_2314(x))
    = (cross-ratio-tuple(x) - 1) / cross-ratio-tuple(x) &
  cross-ratio-tuple(pi_4132(x))
    = (cross-ratio-tuple(x) - 1) / cross-ratio-tuple(x) &
  cross-ratio-tuple(pi_3241(x))
    = (cross-ratio-tuple(x) - 1) / cross-ratio-tuple(x)
  proof
    assume that
A1: x = <* P,Q,R,S *> and
A2: P,Q,R,S are_mutually_distinct and
A3: P,Q,R,S are_collinear;
A4: P <> R & P <> S & R <> Q & S <> Q & S <> R & P <> Q
     by A2,ZFMISC_1:def 6;
    cross-ratio(P,Q,R,S) <> 0 by A3,A4,Th31;
    then reconsider cr = cross-ratio-tuple(x) as non zero Complex by A1,Th36;
    pi_1243(x) = <* P,Q,S,R *> & P,Q,S,R are_collinear &
      P,Q,S,R are_mutually_distinct by A4,A1,A3,ZFMISC_1:def 6;
    then A5: cross-ratio-tuple(pi_1324(pi_1243(x)))
      = 1 - cross-ratio-tuple(pi_1243(x)) by Th41
     .= 1 - (1 / cr) by A1,A3,A4,Th40
     .= cr / cr - 1 / cr by XCMPLX_1:60
     .= (cr - 1) / cr;
    hence cross-ratio-tuple(pi_1423(x))
      = (cross-ratio-tuple(x) - 1) / cross-ratio-tuple(x);
    pi_1423(x) = <* P,S,Q,R *> & P,S,Q,R are_collinear by A1,A3;
    then cross-ratio-tuple(pi_3412(pi_1423(x)))=cross-ratio-tuple(pi_1423(x)) &
      cross-ratio-tuple(pi_2143(pi_1423(x)))=cross-ratio-tuple(pi_1423(x))  &
      cross-ratio-tuple(pi_4321(pi_1423(x)))=cross-ratio-tuple(pi_1423(x))
      by A4,Th37,Th38;
    hence thesis by A5;
  end;

theorem
  x = <* P,Q,R,S *> & P,Q,R,S are_mutually_distinct &
  P,Q,R,S are_collinear implies
  cross-ratio-tuple(pi_1432(x))
    = cross-ratio-tuple(x) / (cross-ratio-tuple(x) - 1) &
  cross-ratio-tuple(pi_2341(x))
    = cross-ratio-tuple(x) / (cross-ratio-tuple(x) - 1) &
  cross-ratio-tuple(pi_3214(x))
    = cross-ratio-tuple(x) / (cross-ratio-tuple(x) - 1) &
  cross-ratio-tuple(pi_4123(x))
    = cross-ratio-tuple(x) / (cross-ratio-tuple(x) - 1)
  proof
    assume that
A1: x = <* P,Q,R,S *> and
A2: P,Q,R,S are_mutually_distinct and
A3: P,Q,R,S are_collinear;
A4: P <> R & P <> S & R <> Q & S <> Q & S <> R & P <> Q
      by A2,ZFMISC_1:def 6;
A5: P,S,R,Q are_collinear & pi_1432(x) = <*P,S,R,Q*> by A1,A3;
A6: cross-ratio-tuple(pi_1432(x))
      = cross-ratio-tuple(pi_1243(pi_1423(x)))
     .= 1 / (cross-ratio-tuple(pi_1423(x))) by Th39
     .= 1 / ((cross-ratio-tuple(x) - 1) / cross-ratio-tuple(x))
       by A1,A3,A2,Th42;
    hence cross-ratio-tuple(pi_1432(x))
      = cross-ratio-tuple(x) / (cross-ratio-tuple(x) - 1) by XCMPLX_1:57;
    now
      thus cross-ratio-tuple(pi_2341(x))
        = cross-ratio-tuple(pi_4321(pi_1432(x)))
       .= cross-ratio-tuple(pi_1432(x)) by A5,A4,Th38;
      thus cross-ratio-tuple(pi_3214(x))
        = cross-ratio-tuple(pi_3412(pi_1432(x)))
       .= cross-ratio-tuple(pi_1432(x)) by A4,A5,Th37;
      thus cross-ratio-tuple(pi_4123(x))
        = cross-ratio-tuple(pi_2143(pi_1432(x)))
       .= cross-ratio-tuple(pi_1432(x)) by A4,A5,Th38;
    end;
    hence thesis by A6,XCMPLX_1:57;
  end;

begin :: Cross-Ratio and real numbers line

Lm03:
  for a,b,c,d being Complex holds
  ((a - c) / (b - c)) / ((a - d) / (b - d)) =
  ((c - a) / (c - b)) * ((d - b) / (d - a))
  proof
    let a,b,c,d be Complex;
    ((a - c) / (b - c)) / ((a - d)/(b - d))
      = (((-1) * (c - a)) /
         (((-1) * (c - b)))) / (((-1) * (d - a))/((-1)*(d - b)))
     .= ((c - a) / (c - b)) / (((-1) * (d - a))/((-1)*(d - b))) by XCMPLX_1:91
     .= ((c - a) / (c - b)) / ((d - a) / (d - b)) by XCMPLX_1:91
     .= ((c - a) / (c - b)) * ((d - b) / (d - a)) by XCMPLX_1:79;
    hence thesis;
  end;

theorem
  for x1,x2,x3,x4 being Element of TOP-REAL 1 st
  x2 <> x3 & x3 <> x1 & x2 <> x4 & x1 <> x4 holds
  ex a,b,c,d being Real st x1 = <* a *> & x2 =  <* b *> & x3 = <* c *> &
  x4 = <* d *> & cross-ratio-tuple(<*x1,x2,x3,x4*>)
    = ((c - a) / (c - b)) * ((d - b) / (d - a))
  proof
    let x1,x2,x3,x4 be Element of TOP-REAL 1;
    assume that
A1: x2 <> x3 and
A2: x3 <> x1 and
A3: x2 <> x4 and
A4: x1 <> x4;
    reconsider x = <*x1,x2,x3,x4*> as Tuple of 4,the carrier of TOP-REAL 1;
    consider P,Q,R,S be Element of TOP-REAL 1 such that
A5: P = x.1 & Q = x.2 & R = x.3 & S = x.4 &
      cross-ratio-tuple(x) = cross-ratio(P,Q,R,S) by Def03;
    consider a1,b1,c1 be Real such that
A7: x3 = <* a1 *> & x1 = <* b1 *> & x2 = <* c1 *> &
    affine-ratio(x3,x1,x2) = (b1 - a1) / (c1 - a1) by A1,A2,Th26;
    consider a2,b2,c2 be Real such that
A8: x4 = <* a2 *> & x1 = <* b2 *> & x2 = <* c2 *> &
    affine-ratio(x4,x1,x2) = (b2 - a2) / (c2 - a2) by A3,A4,Th26;
    take b1,c1,a1,a2;
    b1 = b2 & c1 = c2 by A7,A8,FINSEQ_1:76;
    hence thesis by A7,A8,Lm03,A5;
  end;