:: From Double Loops to Fields
::  by Wojciech Skaba and Micha{\l} Muzalewski

environ

 vocabularies XBOOLE_0, ALGSTR_0, CARD_1, SUPINF_2, VECTSP_1, SUBSET_1,
      RELAT_1, ALGSTR_1, ARYTM_1, ARYTM_3, STRUCT_0, RLVECT_1, BINOP_1,
      LATTICES, MESFUNC1, GROUP_1, ALGSTR_2, NUMBERS;
 notations SUBSET_1, XCMPLX_0, ORDINAL1, NUMBERS, REAL_1, STRUCT_0, ALGSTR_0,
      RLVECT_1, GROUP_1, VECTSP_1, ALGSTR_1;
 constructors BINOP_2, ALGSTR_1, RLVECT_1, VECTSP_1, MEMBERED, REAL_1,
      XXREAL_0, NUMBERS, GROUP_1;
 registrations VECTSP_1, ALGSTR_1, ALGSTR_0, MEMBERED, XREAL_0;
 requirements SUBSET;
 theorems VECTSP_1, ALGSTR_1, RLVECT_1, GROUP_1, STRUCT_0, ALGSTR_0, XCMPLX_1;

begin :: DOUBLE LOOPS

reserve L for non empty doubleLoopStr;

Lm1: 0 = 0.F_Real by STRUCT_0:def 6,VECTSP_1:def 5;

Lm2: for a,b being Element of F_Real st a<>0.F_Real
   ex x being Element of F_Real st a*x=b
proof
  let a,b be Element of F_Real such that
A1: a<>0.F_Real;
  reconsider p=a, q=b as Element of REAL by VECTSP_1:def 5;
  reconsider x = q/p as Element of F_Real by VECTSP_1:def 5;
  p*(q/p) = q by A1,Lm1,XCMPLX_1:87;
  then a*x = b;
  hence thesis;
end;

Lm3: for a,b being Element of F_Real st a<>0.F_Real ex x being Element of
F_Real st x*a=b
proof
  let a,b be Element of F_Real;
  assume a<>0.F_Real;
  then ex x being Element of F_Real st a*x=b by Lm2;
  hence thesis;
end;

Lm4: ( for a,x,y being Element of F_Real st a<>0.F_Real holds a*x=a*y implies
x=y)& for a,x,y being Element of F_Real st a<>0.F_Real holds x*a=y*a implies x=
y by VECTSP_1:5;

Lm5: ( for a being Element of F_Real holds a*0.F_Real = 0.F_Real)& for a being
Element of F_Real holds 0.F_Real*a = 0.F_Real by VECTSP_1:12;

:: Below is the basic definition of the mode of DOUBLE LOOP.
:: The F_Real example in accordance with the many theorems proved above
:: is used to prove the existence.

registration
  cluster F_Real -> multLoop_0-like;
  coherence by Lm2,Lm3,Lm4,Lm5,ALGSTR_1:16;
end;

:: In the following part of this article the negation and minus functions
:: are defined. This is the only definition of both functions in this article
:: while some of their features are independently proved
:: for various structures.

definition
  let L be left_add-cancelable add-right-invertible non empty addLoopStr;
  let a be Element of L;

  func -a -> Element of L means
  :Def1:
  a+it = 0.L;
  existence by ALGSTR_1:def 4;
  uniqueness by ALGSTR_0:def 3;
end;

definition
  let L be left_add-cancelable add-right-invertible non empty addLoopStr;
  let a,b be Element of L;
  func a-b -> Element of L equals
  a+ -b;
  correctness;
end;

registration
  cluster strict Abelian add-associative commutative associative distributive
non degenerated left_zeroed right_zeroed Loop-like well-unital multLoop_0-like
    for non empty doubleLoopStr;
  existence
  proof
    take F_Real;
    thus thesis;
  end;
end;

definition
  mode doubleLoop is left_zeroed right_zeroed Loop-like well-unital
    multLoop_0-like non empty doubleLoopStr;
end;

definition
  mode leftQuasi-Field is Abelian add-associative right-distributive non
    degenerated doubleLoop;
end;

reserve a,b,c,x,y,z for Element of L;

:: The following theorem shows that the basic set of axioms of the
:: left quasi-field may be replaced with the following one,
:: by just removing a few and adding some other axioms.

theorem
  L is leftQuasi-Field iff (for a holds a + 0.L = a) & (for a ex x st a+
x = 0.L) & (for a,b,c holds (a+b)+c = a+(b+c)) & (for a,b holds a+b = b+a) & 0.
L <> 1.L & (for a holds a * 1.L = a) & (for a holds 1.L * a = a) & (for a,b st
a<>0.L ex x st a*x=b) & (for a,b st a<>0.L ex x st x*a=b) & (for a,x,y st a<>0.
L holds a*x=a*y implies x=y) & (for a,x,y st a<>0.L holds x*a=y*a implies x=y)
& (for a holds a*0.L = 0.L) & (for a holds 0.L*a = 0.L) & for a,b,c holds a*(b+
  c) = a*b + a*c
proof
  thus L is leftQuasi-Field implies (for a holds a + 0.L = a) & (for a ex x st
a+x = 0.L) & (for a,b,c holds (a+b)+c = a+(b+c)) & (for a,b holds a+b = b+a) &
  0.L <> 1.L & (for a holds a * 1.L = a) & (for a holds 1.L * a = a) & (for a,b
st a<>0.L ex x st a*x=b) & (for a,b st a<>0.L ex x st x*a=b) & (for a,x,y st a
<>0.L holds a*x=a*y implies x=y) & (for a,x,y st a<>0.L holds x*a=y*a implies x
=y) & (for a holds a*0.L = 0.L) & (for a holds 0.L*a = 0.L) & for a,b,c holds a
*(b+c) = a*b + a*c by ALGSTR_1:6,16,RLVECT_1:def 2,def 3,def 4,STRUCT_0:def 8
,VECTSP_1:def 2;
  assume that
A1: for a holds a + 0.L = a and
A2: for a ex x st a+x = 0.L and
A3: for a,b,c holds (a+b)+c = a+(b+c) and
A4: for a,b holds a+b = b+a and
A5: ( 0.L <> 1.L & for a holds a * 1.L = a )&( ( for a holds 1.L * a = a
  )& for a, b st a<>0.L ex x st a*x=b ) & ( ( for a,b st a<>0.L ex x st x*a=b)&
  for a,x,y st a<>0.L holds a*x=a*y implies x=y )&( ( for a,x,y st a<>0.L holds
x*a=y*a implies x=y)& for a holds a*0.L = 0.L ) &( ( for a holds 0.L*a = 0.L)&
  for a,b,c holds a*(b+c) = a*b + a*c);
A6: for a holds 0.L + a = a
  proof
    let a;
    thus 0.L + a = a + 0.L by A4
      .= a by A1;
  end;
A7: for a,b ex x st a+x=b
  proof
    let a,b;
    consider y such that
A8: a+y = 0.L by A2;
    take x = y+b;
    thus a+x = 0.L + b by A3,A8
      .= b by A6;
  end;
A9: for a,b ex x st x+a=b
  proof
    let a,b;
    consider x such that
A10: a+x=b by A7;
    take x;
    thus thesis by A4,A10;
  end;
A11: for a,x,y holds a+x=a+y implies x=y
  proof
    let a,x,y;
    consider z such that
A12: z+a = 0.L by A1,A2,A3,ALGSTR_1:3;
    assume a+x = a+y;
    then (z+a)+x = z+(a+y) by A3
      .= (z+a)+y by A3;
    hence x = 0.L + y by A6,A12
      .= y by A6;
  end;
  for a,x,y holds x+a=y+a implies x=y
  proof
    let a,x,y;
    assume x+a = y+a;
    then a+x= y+a by A4
      .= a+y by A4;
    hence thesis by A11;
  end;
  hence thesis by A1,A3,A4,A5,A6,A7,A9,A11,ALGSTR_1:6,16,def 2,RLVECT_1:def 2
,def 3,def 4,STRUCT_0:def 8,VECTSP_1:def 2,def 6;
end;

theorem Th2:
  for G being Abelian right-distributive doubleLoop, a,b being
  Element of G holds a*(-b) = -(a*b)
proof
  let G be Abelian right-distributive doubleLoop, a,b be Element of G;
  a*b + a*(-b) = a*(b+ -b) by VECTSP_1:def 2
    .= a*0.G by Def1
    .= 0.G by ALGSTR_1:16;
  hence thesis by Def1;
end;

theorem Th3:
  for G being Abelian left_add-cancelable add-right-invertible
  non empty addLoopStr, a being Element of G holds -(-a) = a
proof
  let G be Abelian left_add-cancelable add-right-invertible non empty
  addLoopStr, a be Element of G;
  -a+a = 0.G by Def1;
  hence thesis by Def1;
end;

theorem
  for G being Abelian right-distributive doubleLoop holds (-1.G)*(-1.G) = 1.G
proof
  let G be Abelian right-distributive doubleLoop;
  thus (-1.G)*(-1.G) = -((-1.G)*1_G) by Th2
    .= -(-1.G)
    .= 1.G by Th3;
end;

theorem
  for G being Abelian right-distributive doubleLoop, a,x,y being Element
  of G holds a*(x-y) = a*x - a*y
proof
  let G be Abelian right-distributive doubleLoop, a,x,y be Element of G;
  thus a*(x-y) = a*x + a*(-y) by VECTSP_1:def 2
    .= a*x - a*y by Th2;
end;

:: RIGHT QUASI-FIELD
:: The next contemplated algebraic structure is so called right quasi-field.
:: This structure is defined as a DOUBLE LOOP augmented with three axioms.
:: The reasoning is similar to that of left quasi-field.

definition
  mode rightQuasi-Field is Abelian add-associative left-distributive non
    degenerated doubleLoop;
end;

theorem
  L is rightQuasi-Field iff (for a holds a + 0.L = a) & (for a ex x st a
  +x = 0.L) & (for a,b,c holds (a+b)+c = a+(b+c)) & (for a,b holds a+b = b+a) &
  0.L <> 1.L & (for a holds a * 1.L = a) & (for a holds 1.L * a = a) & (for a,b
st a<>0.L ex x st a*x=b) & (for a,b st a<>0.L ex x st x*a=b) & (for a,x,y st a
<>0.L holds a*x=a*y implies x=y) & (for a,x,y st a<>0.L holds x*a=y*a implies x
=y) & (for a holds a*0.L = 0.L) & (for a holds 0.L*a = 0.L) & for a,b,c holds (
  b+c)*a = b*a + c*a
proof
  thus L is rightQuasi-Field implies (for a holds a + 0.L = a) & (for a ex x
st a+x = 0.L) & (for a,b,c holds (a+b)+c = a+(b+c)) & (for a,b holds a+b = b+a)
& 0.L <> 1.L & (for a holds a * 1.L = a) & (for a holds 1.L * a = a) & (for a,b
st a<>0.L ex x st a*x=b) & (for a,b st a<>0.L ex x st x*a=b) & (for a,x,y st a
<>0.L holds a*x=a*y implies x=y) & (for a,x,y st a<>0.L holds x*a=y*a implies x
=y) & (for a holds a*0.L = 0.L) & (for a holds 0.L*a = 0.L) & for a,b,c holds (
b+c)*a = b*a + c*a by ALGSTR_1:6,16,RLVECT_1:def 2,def 3,def 4,STRUCT_0:def 8
,VECTSP_1:def 3;
  assume that
A1: for a holds a + 0.L = a and
A2: for a ex x st a+x = 0.L and
A3: for a,b,c holds (a+b)+c = a+(b+c) and
A4: for a,b holds a+b = b+a and
A5: ( 0.L <> 1.L & for a holds a * 1.L = a )&( ( for a holds 1.L * a = a
  )& for a, b st a<>0.L ex x st a*x=b ) & ( ( for a,b st a<>0.L ex x st x*a=b)&
  for a,x,y st a<>0.L holds a*x=a*y implies x=y )&( ( for a,x,y st a<>0.L holds
x*a=y*a implies x=y)& for a holds a*0.L = 0.L ) &( ( for a holds 0.L*a = 0.L)&
  for a,b,c holds (b+c)*a = b*a + c*a);
A6: for a,b ex x st x+a=b
  proof
    let a,b;
    consider y such that
A7: y+a = 0.L by A1,A2,A3,ALGSTR_1:3;
    take x = b+y;
    thus x+a = b + 0.L by A3,A7
      .= b by A1;
  end;
A8: for a holds 0.L + a = a
  proof
    let a;
    thus 0.L + a = a + 0.L by A4
      .= a by A1;
  end;
A9: for a,x,y holds a+x=a+y implies x=y
  proof
    let a,x,y;
    consider z such that
A10: z+a = 0.L by A1,A2,A3,ALGSTR_1:3;
    assume a+x = a+y;
    then (z+a)+x = z+(a+y) by A3
      .= (z+a)+y by A3;
    hence x = 0.L + y by A8,A10
      .= y by A8;
  end;
A11: for a,x,y holds x+a=y+a implies x=y
  proof
    let a,x,y;
    consider z such that
A12: a+z = 0.L by A2;
    assume x+a = y+a;
    then x+(a+z) = (y+a)+z by A3
      .= y+(a+z) by A3;
    hence x = y + 0.L by A1,A12
      .= y by A1;
  end;
  for a,b ex x st a+x=b
  proof
    let a,b;
    consider y such that
A13: a+y = 0.L by A2;
    take x = y+b;
    thus a+x = 0.L + b by A3,A13
      .= b by A8;
  end;
  hence thesis by A1,A3,A4,A5,A8,A6,A9,A11,ALGSTR_1:6,16,def 2,RLVECT_1:def 2
,def 3,def 4,STRUCT_0:def 8,VECTSP_1:def 3,def 6;
end;

:: Below, the three features concerned with the - function,
:: numbered 20..22 are proved. Where necessary, a few additional
:: facts are included. They are independent of the similar proofs
:: performed for the left quasi-field.

reserve G for left-distributive doubleLoop,
  a,b,x,y for Element of G;

theorem Th7:
  (-b)*a = -(b*a)
proof
  b*a + (-b)*a = (b+(-b))*a by VECTSP_1:def 3
    .= 0.G*a by Def1
    .= 0.G by ALGSTR_1:16;
  hence thesis by Def1;
end;

theorem
  for G being Abelian left-distributive doubleLoop holds (-1.G)*(-1.G) = 1.G
proof
  let G be Abelian left-distributive doubleLoop;
  thus (-1.G)*(-1.G) = -(1_G*(-1.G)) by Th7
    .= -(-1.G)
    .= 1.G by Th3;
end;

theorem
  (x-y)*a = x*a - y*a
proof
  thus (x-y)*a = x*a + (-y)*a by VECTSP_1:def 3
    .= x*a - y*a by Th7;
end;

:: DOUBLE SIDED QUASI-FIELD
:: The next contemplated algebraic structure is so called double sided
:: quasi-field. This structure is also defined as a DOUBLE LOOP augmented
:: with four axioms, while its relevance to left/right quasi-field is
:: independently contemplated.
:: The reasoning is similar to that of left/right quasi-field.

definition
  mode doublesidedQuasi-Field is Abelian add-associative distributive non
    degenerated doubleLoop;
end;

reserve a,b,c,x,y,z for Element of L;

theorem
  L is doublesidedQuasi-Field iff (for a holds a + 0.L = a) & (for a ex
x st a+x = 0.L) & (for a,b,c holds (a+b)+c = a+(b+c)) & (for a,b holds a+b = b+
a) & 0.L <> 1.L & (for a holds a * 1.L = a) & (for a holds 1.L * a = a) & (for
  a,b st a<>0.L ex x st a*x=b) & (for a,b st a<>0.L ex x st x*a=b) & (for a,x,y
  st a<>0.L holds a*x=a*y implies x=y) & (for a,x,y st a<>0.L holds x*a=y*a
implies x=y) & (for a holds a*0.L = 0.L) & (for a holds 0.L*a = 0.L) & (for a,b
  ,c holds a*(b+c) = a*b + a*c) & for a,b,c holds (b+c)*a = b*a + c*a
proof
  thus L is doublesidedQuasi-Field implies (for a holds a + 0.L = a) & (for a
ex x st a+x = 0.L) & (for a,b,c holds (a+b)+c = a+(b+c)) & (for a,b holds a+b =
  b+a) & 0.L <> 1.L & (for a holds a * 1.L = a) & (for a holds 1.L * a = a) & (
for a,b st a<>0.L ex x st a*x=b) & (for a,b st a<>0.L ex x st x*a=b) & (for a,x
  ,y st a<>0.L holds a*x=a*y implies x=y) & (for a,x,y st a<>0.L holds x*a=y*a
implies x=y) & (for a holds a*0.L = 0.L) & (for a holds 0.L*a = 0.L) & (for a,b
,c holds a*(b+c) = a*b + a*c) & for a,b,c holds (b+c)*a = b*a + c*a by
ALGSTR_1:6,16,RLVECT_1:def 2,def 3,def 4,STRUCT_0:def 8,VECTSP_1:def 7;
  assume that
A1: for a holds a + 0.L = a and
A2: for a ex x st a+x = 0.L and
A3: for a,b,c holds (a+b)+c = a+(b+c) and
A4: for a,b holds a+b = b+a and
A5: ( 0.L <> 1.L & for a holds a * 1.L = a )&( ( for a holds 1.L * a = a
  )& for a, b st a<>0.L ex x st a*x=b ) & ( ( for a,b st a<>0.L ex x st x*a=b)&
  for a,x,y st a<>0.L holds a*x=a*y implies x=y )&( ( for a,x,y st a<>0.L holds
x*a=y*a implies x=y)& for a holds a*0.L = 0.L ) &( ( ( for a holds 0.L*a = 0.L)
& for a,b,c holds a*(b+c) = a*b + a*c )& for a,b, c holds (b+c)*a = b*a + c*a);
A6: for a,b ex x st x+a=b
  proof
    let a,b;
    consider y such that
A7: y+a = 0.L by A1,A2,A3,ALGSTR_1:3;
    take x = b+y;
    thus x+a = b + 0.L by A3,A7
      .= b by A1;
  end;
A8: for a holds 0.L + a = a
  proof
    let a;
    thus 0.L + a = a + 0.L by A4
      .= a by A1;
  end;
A9: for a,x,y holds a+x=a+y implies x=y
  proof
    let a,x,y;
    consider z such that
A10: z+a = 0.L by A1,A2,A3,ALGSTR_1:3;
    assume a+x = a+y;
    then (z+a)+x = z+(a+y) by A3
      .= (z+a)+y by A3;
    hence x = 0.L + y by A8,A10
      .= y by A8;
  end;
A11: for a,x,y holds x+a=y+a implies x=y
  proof
    let a,x,y;
    consider z such that
A12: a+z = 0.L by A2;
    assume x+a = y+a;
    then x+(a+z) = (y+a)+z by A3
      .= y+(a+z) by A3;
    hence x = y + 0.L by A1,A12
      .= y by A1;
  end;
  for a,b ex x st a+x=b
  proof
    let a,b;
    consider y such that
A13: a+y = 0.L by A2;
    take x = y+b;
    thus a+x = 0.L + b by A3,A13
      .= b by A8;
  end;
  hence thesis by A1,A3,A4,A5,A8,A6,A9,A11,ALGSTR_1:6,16,def 2,RLVECT_1:def 2
,def 3,def 4,STRUCT_0:def 8,VECTSP_1:def 6,def 7;
end;

:: SKEW FIELD
:: A Skew-Field is defined as a double sided quasi-field extended
:: with the associativity of multiplication.

definition
  mode _Skew-Field is associative doublesidedQuasi-Field;
end;

Lm6: 0.L <> 1.L & (for a holds a * 1.L = a) & (for a st a<>0.L ex x st a*x =
1.L) & (for a,b,c holds (a*b)*c = a*(b*c)) & (for a holds a*0.L = 0.L) implies
(a*b = 1.L implies b*a = 1.L)
proof
  assume that
A1: 0.L <> 1.L and
A2: for a holds a * 1.L = a and
A3: for a st a<>0.L ex x st a*x = 1.L and
A4: for a,b,c holds (a*b)*c = a*(b*c) and
A5: for a holds a*0.L = 0.L;
  thus a*b = 1.L implies b*a = 1.L
  proof
    assume
A6: a*b = 1.L;
    then b<>0.L by A1,A5;
    then consider x such that
A7: b * x = 1.L by A3;
    thus b*a = (b*a) * (b*x) by A2,A7
      .= ((b*a) * b) * x by A4
      .= (b * 1.L) * x by A4,A6
      .= 1.L by A2,A7;
  end;
end;

Lm7: 0.L <> 1.L & (for a holds a * 1.L = a) & (for a st a<>0.L ex x st a*x =
1.L) & (for a,b,c holds (a*b)*c = a*(b*c)) & (for a holds a*0.L = 0.L) implies
1.L*a = a*1.L
proof
  assume that
A1: 0.L <> 1.L and
A2: for a holds a * 1.L = a and
A3: for a st a<>0.L ex x st a*x = 1.L and
A4: for a,b,c holds (a*b)*c = a*(b*c) and
A5: for a holds a*0.L = 0.L;
A6: a<>0.L implies 1.L*a = a*1.L
  proof
    assume a<>0.L;
    then consider x such that
A7: a * x = 1.L by A3;
    thus 1.L*a = a * (x*a) by A4,A7
      .= a*1.L by A1,A2,A3,A4,A5,A7,Lm6;
  end;
  a=0.L implies 1.L*a = a*1.L
  proof
    assume
A8: a=0.L;
    hence 1.L*a = 0.L by A5
      .= a*1.L by A2,A8;
  end;
  hence thesis by A6;
end;

Lm8: 0.L <> 1.L & (for a holds a * 1.L = a) & (for a st a<>0.L ex x st a*x =
1.L) & (for a,b,c holds (a*b)*c = a*(b*c)) & (for a holds a*0.L = 0.L) implies
for a st a<>0.L ex x st x*a = 1.L
proof
  assume that
A1: 0.L <> 1.L & for a holds a * 1.L = a and
A2: for a st a<>0.L ex x st a*x = 1.L and
A3: ( for a,b,c holds (a*b)*c = a*(b*c))& for a holds a*0.L = 0.L;
  let a;
  assume a<>0.L;
  then consider x such that
A4: a * x = 1.L by A2;
  x*a=1.L by A1,A2,A3,A4,Lm6;
  hence thesis;
end;

:: The following theorem shows that the basic set of axioms of the
:: skew field may be replaced with the following one,
:: by just removing a few and adding some other axioms.
:: A few theorems proved earlier are highly utilized.

theorem Th11:
  L is _Skew-Field iff (for a holds a + 0.L = a) & (for a ex x st a+x = 0.L)
  & (for a,b,c holds (a+b)+c = a+(b+c)) & (for a,b holds a+b = b+a)
  & 0.L <> 1.L & (for a holds a * 1.L = a)
  & (for a st a<>0.L ex x st a*x = 1.L) & (for a holds a*0.L = 0.L)
  & (for a holds 0.L*a = 0.L) & (for a,b,c holds (a*b)*c = a*(b*c))
  & (for a,b,c holds a*(b+c) = a*b + a*c)
  & (for a,b,c holds (b+c)*a = b*a + c*a)
proof
  thus L is _Skew-Field implies (for a holds a + 0.L = a)
  & (for a ex x st a+x = 0.L) & (for a,b,c holds (a+b)+c = a+(b+c))
  & (for a,b holds a+b = b+a) & 0.L <> 1.L & (for a holds a * 1.L = a)
  & (for a st a<>0.L ex x st a*x = 1.L) & (for a holds a*0.L = 0.L)
  & (for a holds 0.L*a = 0.L) & (for a,b,c holds (a*b)*c = a*(b*c))
  & (for a,b,c holds a*(b+c) = a*b + a*c)
  & (for a,b,c holds (b+c)*a = b*a + c*a) by ALGSTR_1:6,16,GROUP_1:def 3
,RLVECT_1:def 2,def 3,def 4,STRUCT_0:def 8,VECTSP_1:def 7;
  assume
A1: (for a holds a + 0.L = a) & (for a ex x st a+x = 0.L)
  & (for a,b,c holds (a+b)+c = a+(b+c)) & (for a,b holds a+b = b+a)
  & 0.L <> 1.L & (for a holds a * 1.L = a)
  & (for a st a<>0.L ex x st a*x = 1.L) & (for a holds a*0.L = 0.L)
  & (for a holds 0.L*a = 0.L) & (for a,b,c holds (a*b)*c = a*(b*c))
  & (for a,b,c holds a*(b+c) = a*b + a*c)
  & (for a,b,c holds (b+c)*a = b*a + c*a);
  now thus
A2: for a holds 0.L + a = a
    proof
      let a;
      thus 0.L + a = a + 0.L by A1
        .= a by A1;
    end;
    thus for a,b ex x st a+x=b
    proof
      let a,b;
      consider y such that
A3:   a+y = 0.L by A1;
      take x = y+b;
      thus a+x = 0.L + b by A1,A3
        .= b by A2;
    end;
    thus for a,b ex x st x+a=b
    proof
      let a,b;
      consider y such that
A4:   y+a = 0.L by A1,ALGSTR_1:3;
      take x = b+y;
      thus x+a = b + 0.L by A1,A4
        .= b by A1;
    end;
    thus for a,x,y holds a+x=a+y implies x=y
    proof
      let a,x,y;
      consider z such that
A5:   z+a = 0.L by A1,ALGSTR_1:3;
      assume a+x = a+y;
      then (z+a)+x = z+(a+y) by A1
        .= (z+a)+y by A1;
      hence x = 0.L + y by A2,A5
        .= y by A2;
    end;
    thus for a,x,y holds x+a=y+a implies x=y
    proof
      let a,x,y;
      consider z such that
A6:   a+z = 0.L by A1;
      assume x+a = y+a;
      then x+(a+z) = (y+a)+z by A1
        .= y+(a+z) by A1;
      hence x = y + 0.L by A1,A6
        .= y by A1;
    end; thus
A7: for a holds 1.L * a = a
    proof
      let a;
      thus 1.L*a = a*1.L by A1,Lm7
        .= a by A1;
    end;
    thus for a,b st a<>0.L ex x st a*x=b
    proof
      let a,b;
      assume a<>0.L;
      then consider y such that
A8:   a*y = 1.L by A1;
      take x = y*b;
      thus a*x = 1.L * b by A1,A8
        .= b by A7;
    end;
    thus for a,b st a<>0.L ex x st x*a=b
    proof
      let a,b;
      assume a<>0.L;
      then consider y such that
A9:   y*a = 1.L by A1,Lm8;
      take x = b*y;
      thus x*a = b * 1.L by A1,A9
        .= b by A1;
    end;
    thus for a,x,y st a<>0.L holds a*x=a*y implies x=y
    proof
      let a,x,y;
      assume a<>0.L;
      then consider z such that
A10:  z*a = 1.L by A1,Lm8;
      assume a*x = a*y;
      then (z*a)*x = z*(a*y) by A1
        .= (z*a)*y by A1;
      hence x = 1.L * y by A7,A10
        .= y by A7;
    end;
    thus for a,x,y st a<>0.L holds x*a=y*a implies x=y
    proof
      let a,x,y;
      assume a<>0.L;
      then consider z such that
A11:  a*z = 1.L by A1;
      assume x*a = y*a;
      then x*(a*z) = (y*a)*z by A1
        .= y*(a*z) by A1;
      hence x = y * 1.L by A1,A11
        .= y by A1;
    end;
  end;
  hence thesis by A1,ALGSTR_1:6,16,def 2,GROUP_1:def 3,RLVECT_1:def 2,def 3
,def 4,STRUCT_0:def 8,VECTSP_1:def 6,def 7;
end;

:: FIELD
:: A _Field is defined as a Skew-Field with the axiom of the commutativity
:: of multiplication.

definition
  mode _Field is commutative _Skew-Field;
end;

theorem
  L is _Field iff (for a holds a + 0.L = a) & (for a ex x st a+x = 0.L)
& (for a,b,c holds (a+b)+c = a+(b+c)) & (for a,b holds a+b = b+a) & 0.L <> 1.L
  & (for a holds a * 1.L = a) & (for a st a<>0.L ex x st a*x = 1.L) & (for a
holds a*0.L = 0.L) & (for a,b,c holds (a*b)*c = a*(b*c)) & (for a,b,c holds a*(
  b+c) = a*b + a*c) & for a,b holds a*b = b*a
proof
  thus L is _Field implies (for a holds a + 0.L = a) & (for a ex x st a+x = 0.
  L) & (for a,b,c holds (a+b)+c = a+(b+c)) & (for a,b holds a+b = b+a) & 0.L <>
1.L & (for a holds a * 1.L = a) & (for a st a<>0.L ex x st a*x = 1.L) & (for a
holds a*0.L = 0.L) & (for a,b,c holds (a*b)*c = a*(b*c)) & (for a,b,c holds a*(
  b+c) = a*b + a*c) & for a,b holds a*b = b*a by Th11,GROUP_1:def 12;
  assume that
A1: ( ( for a holds a + 0.L = a)& for a ex x st a+x = 0.L )&( ( for a,b,
c holds ( a +b)+c = a+(b+c))& for a,b holds a+b = b+a ) &( ( 0.L <> 1.L & for a
  holds a * 1.L = a )& for a st a<>0.L ex x st a*x = 1.L ) and
A2: for a holds a*0.L = 0.L and
A3: for a,b,c holds (a*b)*c = a*(b*c) and
A4: for a,b,c holds a*(b+c) = a*b + a*c and
A5: for a,b holds a*b = b*a;
A6: for a holds 0.L*a = 0.L
  proof
    let a;
    thus 0.L*a = a*0.L by A5
      .= 0.L by A2;
  end;
  for a,b,c holds (b+c)*a = b*a + c*a
  proof
    let a,b,c;
    thus (b+c)*a = a*(b+c) by A5
      .= a*b + a*c by A4
      .= b*a + a*c by A5
      .= b*a + c*a by A5;
  end;
  hence thesis by A1,A2,A3,A4,A5,A6,Th11,GROUP_1:def 12;
end;