Article REALSET2, MML version 4.99.1005

:: REALSET2:funcnot 1 => REALSET2:func 1
definition
  func add_2 -> Function-like quasi_total Relation of [:2,2:],2 equals
    ((((0,0).--> 0) +* ((0,1).--> 1)) +* ((1,0).--> 1)) +* ((1,1).--> 0);
end;

:: REALSET2:def 8
theorem
add_2 = ((((0,0).--> 0) +* ((0,1).--> 1)) +* ((1,0).--> 1)) +* ((1,1).--> 0);

:: REALSET2:funcnot 2 => REALSET2:func 2
definition
  func mult_2 -> Function-like quasi_total Relation of [:2,2:],2 equals
    ((((0,0).--> 0) +* ((0,1).--> 0)) +* ((1,0).--> 0)) +* ((1,1).--> 1);
end;

:: REALSET2:def 9
theorem
mult_2 = ((((0,0).--> 0) +* ((0,1).--> 0)) +* ((1,0).--> 0)) +* ((1,1).--> 1);

:: REALSET2:funcnot 3 => REALSET2:func 3
definition
  func dL-Z_2 -> doubleLoopStr equals
    doubleLoopStr(#2,add_2,mult_2,In(1,2),In(0,2)#);
end;

:: REALSET2:def 10
theorem
dL-Z_2 = doubleLoopStr(#2,add_2,mult_2,In(1,2),In(0,2)#);

:: REALSET2:funcreg 1
registration
  cluster dL-Z_2 -> non empty non degenerated strict;
end;

:: REALSET2:funcreg 2
registration
  cluster dL-Z_2 -> add-associative distributive;
end;

:: REALSET2:exreg 1
registration
  cluster non empty non trivial strict add-associative doubleLoopStr;
end;

:: REALSET2:condreg 1
registration
  cluster -> natural (Element of the carrier of dL-Z_2);
end;

:: REALSET2:exreg 2
registration
  cluster empty epsilon-transitive epsilon-connected ordinal natural Element of the carrier of dL-Z_2;
end;

:: REALSET2:exreg 3
registration
  cluster non empty epsilon-transitive epsilon-connected ordinal natural Element of the carrier of dL-Z_2;
end;

:: REALSET2:attrnot 1 => REALSET2:attr 1
definition
  let a1 be non trivial doubleLoopStr;
  attr a1 is Field-like means
    the multF of a1 is DnT of 0. a1,the carrier of a1 &
     (for b1 being non empty set
     for b2 being Function-like quasi_total Relation of [:b1,b1:],b1
     for b3 being Element of b1
           st b1 = (the carrier of a1) \ {0. a1} &
              b3 = 1. a1 &
              b2 = (the multF of a1) || ((the carrier of a1) \ {0. a1})
        holds addLoopStr(#b1,b2,b3#) is non empty right_complementable Abelian add-associative right_zeroed addLoopStr);
end;

:: REALSET2:dfs 4
definiens
  let a1 be non trivial doubleLoopStr;
To prove
     a1 is Field-like
it is sufficient to prove
  thus the multF of a1 is DnT of 0. a1,the carrier of a1 &
     (for b1 being non empty set
     for b2 being Function-like quasi_total Relation of [:b1,b1:],b1
     for b3 being Element of b1
           st b1 = (the carrier of a1) \ {0. a1} &
              b3 = 1. a1 &
              b2 = (the multF of a1) || ((the carrier of a1) \ {0. a1})
        holds addLoopStr(#b1,b2,b3#) is non empty right_complementable Abelian add-associative right_zeroed addLoopStr);

:: REALSET2:def 11
theorem
for b1 being non trivial doubleLoopStr holds
      b1 is Field-like
   iff
      the multF of b1 is DnT of 0. b1,the carrier of b1 &
       (for b2 being non empty set
       for b3 being Function-like quasi_total Relation of [:b2,b2:],b2
       for b4 being Element of b2
             st b2 = (the carrier of b1) \ {0. b1} &
                b4 = 1. b1 &
                b3 = (the multF of b1) || ((the carrier of b1) \ {0. b1})
          holds addLoopStr(#b2,b3,b4#) is non empty right_complementable Abelian add-associative right_zeroed addLoopStr);

:: REALSET2:funcreg 3
registration
  let a1 be non trivial set;
  let a2 be Function-like quasi_total Relation of [:a1,a1:],a1;
  let a3 be Element of a1;
  let a4 be Function-like quasi_total Relation of [:a1,a1:],a1;
  let a5 be Element of a1;
  cluster doubleLoopStr(#a1,a2,a4,a5,a3#) -> non empty strict;
end;

:: REALSET2:condreg 2
registration
  cluster non trivial -> non empty (doubleLoopStr);
end;

:: REALSET2:exreg 4
registration
  cluster non empty non degenerated non trivial right_complementable strict Abelian add-associative right_zeroed distributive Field-like doubleLoopStr;
end;

:: REALSET2:condreg 3
registration
  cluster non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive -> Field-like (doubleLoopStr);
end;

:: REALSET2:funcnot 4 => REALSET2:func 4
definition
  let a1 be non degenerated right_complementable Abelian add-associative right_zeroed distributive Field-like doubleLoopStr;
  func omf A1 -> DnT of 0. a1,the carrier of a1 equals
    the multF of a1;
end;

:: REALSET2:def 15
theorem
for b1 being non degenerated right_complementable Abelian add-associative right_zeroed distributive Field-like doubleLoopStr holds
   omf b1 = the multF of b1;

:: REALSET2:th 1
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr holds
   addLoopStr(#the carrier of b1,the addF of b1,the ZeroF of b1#) is non empty right_complementable Abelian add-associative right_zeroed addLoopStr;

:: REALSET2:th 6
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being Element of the carrier of b1 holds
   b2 + 0. b1 = b2 & (0. b1) + b2 = b2;

:: REALSET2:th 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for b2 being Element of the carrier of b1 holds
   ex b3 being Element of the carrier of b1 st
      b2 + b3 = 0. b1 & b3 + b2 = 0. b1;

:: REALSET2:th 8
theorem
for b1 being non degenerated right_complementable Abelian add-associative right_zeroed distributive Field-like doubleLoopStr
for b2, b3, b4 being Element of (the carrier of b1) \ {0. b1} holds
(b2 * b3) * b4 = b2 * (b3 * b4);

:: REALSET2:th 9
theorem
for b1 being non degenerated right_complementable Abelian add-associative right_zeroed distributive Field-like doubleLoopStr
for b2, b3 being Element of (the carrier of b1) \ {0. b1} holds
b2 * b3 = b3 * b2;

:: REALSET2:th 10
theorem
for b1 being non degenerated right_complementable Abelian add-associative right_zeroed distributive Field-like doubleLoopStr
for b2 being Element of (the carrier of b1) \ {0. b1} holds
   b2 * 1. b1 = b2 & (1. b1) * b2 = b2;

:: REALSET2:th 11
theorem
for b1 being non degenerated right_complementable Abelian add-associative right_zeroed distributive Field-like doubleLoopStr
for b2 being Element of (the carrier of b1) \ {0. b1} holds
   ex b3 being Element of (the carrier of b1) \ {0. b1} st
      b2 * b3 = 1. b1 & b3 * b2 = 1. b1;

:: REALSET2:th 12
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3 being Element of the carrier of b1
      st b2 + b3 = 0. b1
   holds b3 = (comp b1) . b2;

:: REALSET2:th 13
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being Element of the carrier of b1 holds
   b2 = (comp b1) . ((comp b1) . b2);

:: REALSET2:th 14
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3 being Element of the carrier of b1 holds
(the addF of b1) .(b2,b3) is Element of the carrier of b1 & (omf b1) .(b2,b3) is Element of the carrier of b1 & (comp b1) . b2 is Element of the carrier of b1;

:: REALSET2:th 15
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3, b4 being Element of the carrier of b1 holds
b2 * (b3 - b4) = (b2 * b3) - (b2 * b4);

:: REALSET2:th 16
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 - b3) * b4 = (b2 * b4) - (b3 * b4);

:: REALSET2:th 17
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being Element of the carrier of b1 holds
   b2 * 0. b1 = 0. b1;

:: REALSET2:th 18
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being Element of the carrier of b1 holds
   (0. b1) * b2 = 0. b1;

:: REALSET2:th 19
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3 being Element of the carrier of b1 holds
- (b2 * b3) = b2 * - b3;

:: REALSET2:th 20
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr holds
   (1. b1) * 0. b1 = 0. b1;

:: REALSET2:th 21
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr holds
   (0. b1) * 1. b1 = 0. b1;

:: REALSET2:th 22
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3 being Element of the carrier of b1 holds
(omf b1) .(b2,b3) is Element of the carrier of b1;

:: REALSET2:th 23
theorem
for b1 being non degenerated right_complementable Abelian add-associative right_zeroed distributive Field-like doubleLoopStr
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 * b3) * b4 = b2 * (b3 * b4);

:: REALSET2:th 24
theorem
for b1 being non degenerated right_complementable Abelian add-associative right_zeroed distributive Field-like doubleLoopStr
for b2, b3 being Element of the carrier of b1 holds
b2 * b3 = b3 * b2;

:: REALSET2:th 25
theorem
for b1 being non degenerated right_complementable Abelian add-associative right_zeroed distributive Field-like doubleLoopStr
for b2 being Element of the carrier of b1 holds
   b2 * 1. b1 = b2 & (1. b1) * b2 = b2;

:: REALSET2:funcnot 5 => REALSET2:func 5
definition
  let a1 be non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
  func revf A1 -> Function-like quasi_total Relation of (the carrier of a1) \ {0. a1},(the carrier of a1) \ {0. a1} means
    for b1 being Element of (the carrier of a1) \ {0. a1} holds
       (omf a1) .(b1,it . b1) = 1. a1;
end;

:: REALSET2:def 18
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being Function-like quasi_total Relation of (the carrier of b1) \ {0. b1},(the carrier of b1) \ {0. b1} holds
      b2 = revf b1
   iff
      for b3 being Element of (the carrier of b1) \ {0. b1} holds
         (omf b1) .(b3,b2 . b3) = 1. b1;

:: REALSET2:th 26
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3 being Element of (the carrier of b1) \ {0. b1}
      st b2 * b3 = 1. b1
   holds b3 = (revf b1) . b2;

:: REALSET2:th 27
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being Element of (the carrier of b1) \ {0. b1} holds
   b2 = (revf b1) . ((revf b1) . b2);

:: REALSET2:th 28
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3 being Element of (the carrier of b1) \ {0. b1} holds
(omf b1) .(b2,b3) is Element of (the carrier of b1) \ {0. b1} &
 (revf b1) . b2 is Element of (the carrier of b1) \ {0. b1};

:: REALSET2:th 29
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3, b4 being Element of the carrier of b1
      st b2 + b3 = b2 + b4
   holds b3 = b4;

:: REALSET2:th 30
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being Element of (the carrier of b1) \ {0. b1}
for b3, b4 being Element of the carrier of b1
      st b2 * b3 = b2 * b4
   holds b3 = b4;

:: REALSET2:condreg 4
registration
  cluster non degenerated right_complementable Abelian add-associative right_zeroed distributive Field-like -> almost_left_invertible associative commutative well-unital (doubleLoopStr);
end;

:: REALSET2:th 31
theorem
for b1 being non degenerated doubleLoopStr holds
   1. b1 is Element of (the carrier of b1) \ {0. b1};