Article PARSP_1, MML version 4.99.1005

:: PARSP_1:funcnot 1 => PARSP_1:func 1
definition
  let a1 be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
  func c3add A1 -> Function-like quasi_total Relation of [:[:the carrier of a1,the carrier of a1,the carrier of a1:],[:the carrier of a1,the carrier of a1,the carrier of a1:]:],[:the carrier of a1,the carrier of a1,the carrier of a1:] means
    for b1, b2 being Element of [:the carrier of a1,the carrier of a1,the carrier of a1:] holds
    it .(b1,b2) = [b1 `1 + (b2 `1),b1 `2 + (b2 `2),b1 `3 + (b2 `3)];
end;

:: PARSP_1:def 1
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being Function-like quasi_total Relation of [:[:the carrier of b1,the carrier of b1,the carrier of b1:],[:the carrier of b1,the carrier of b1,the carrier of b1:]:],[:the carrier of b1,the carrier of b1,the carrier of b1:] holds
      b2 = c3add b1
   iff
      for b3, b4 being Element of [:the carrier of b1,the carrier of b1,the carrier of b1:] holds
      b2 .(b3,b4) = [b3 `1 + (b4 `1),b3 `2 + (b4 `2),b3 `3 + (b4 `3)];

:: PARSP_1:funcnot 2 => PARSP_1:func 2
definition
  let a1 be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
  let a2, a3 be Element of [:the carrier of a1,the carrier of a1,the carrier of a1:];
  func A2 + A3 -> Element of [:the carrier of a1,the carrier of a1,the carrier of a1:] equals
    (c3add a1) .(a2,a3);
end;

:: PARSP_1:def 2
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2, b3 being Element of [:the carrier of b1,the carrier of b1,the carrier of b1:] holds
b2 + b3 = (c3add b1) .(b2,b3);

:: PARSP_1:th 3
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2, b3 being Element of [:the carrier of b1,the carrier of b1,the carrier of b1:] holds
b2 + b3 = [b2 `1 + (b3 `1),b2 `2 + (b3 `2),b2 `3 + (b3 `3)];

:: PARSP_1:th 4
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1 holds
[b2,b3,b4] + [b5,b6,b7] = [b2 + b5,b3 + b6,b4 + b7];

:: PARSP_1:funcnot 3 => PARSP_1:func 3
definition
  let a1 be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
  func c3compl A1 -> Function-like quasi_total Relation of [:the carrier of a1,the carrier of a1,the carrier of a1:],[:the carrier of a1,the carrier of a1,the carrier of a1:] means
    for b1 being Element of [:the carrier of a1,the carrier of a1,the carrier of a1:] holds
       it . b1 = [- (b1 `1),- (b1 `2),- (b1 `3)];
end;

:: PARSP_1:def 3
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being Function-like quasi_total Relation of [:the carrier of b1,the carrier of b1,the carrier of b1:],[:the carrier of b1,the carrier of b1,the carrier of b1:] holds
      b2 = c3compl b1
   iff
      for b3 being Element of [:the carrier of b1,the carrier of b1,the carrier of b1:] holds
         b2 . b3 = [- (b3 `1),- (b3 `2),- (b3 `3)];

:: PARSP_1:funcnot 4 => PARSP_1:func 4
definition
  let a1 be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
  let a2 be Element of [:the carrier of a1,the carrier of a1,the carrier of a1:];
  func - A2 -> Element of [:the carrier of a1,the carrier of a1,the carrier of a1:] equals
    (c3compl a1) . a2;
end;

:: PARSP_1:def 4
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being Element of [:the carrier of b1,the carrier of b1,the carrier of b1:] holds
   - b2 = (c3compl b1) . b2;

:: PARSP_1:th 7
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being Element of [:the carrier of b1,the carrier of b1,the carrier of b1:] holds
   - b2 = [- (b2 `1),- (b2 `2),- (b2 `3)];

:: PARSP_1:modenot 1 => PARSP_1:mode 1
definition
  let a1 be set;
  mode Relation4 of A1 means
    it c= [:a1,a1,a1,a1:];
end;

:: PARSP_1:dfs 5
definiens
  let a1, a2 be set;
To prove
     a2 is Relation4 of a1
it is sufficient to prove
  thus a2 c= [:a1,a1,a1,a1:];

:: PARSP_1:def 5
theorem
for b1, b2 being set holds
   b2 is Relation4 of b1
iff
   b2 c= [:b1,b1,b1,b1:];

:: PARSP_1:structnot 1 => PARSP_1:struct 1
definition
  struct(1-sorted) ParStr(#
    carrier -> set,
    4_arg_relation -> Relation4 of the carrier of it
  #);
end;

:: PARSP_1:attrnot 1 => PARSP_1:attr 1
definition
  let a1 be ParStr;
  attr a1 is strict;
end;

:: PARSP_1:exreg 1
registration
  cluster strict ParStr;
end;

:: PARSP_1:aggrnot 1 => PARSP_1:aggr 1
definition
  let a1 be set;
  let a2 be Relation4 of a1;
  aggr ParStr(#a1,a2#) -> strict ParStr;
end;

:: PARSP_1:selnot 1 => PARSP_1:sel 1
definition
  let a1 be ParStr;
  sel the 4_arg_relation of a1 -> Relation4 of the carrier of a1;
end;

:: PARSP_1:exreg 2
registration
  cluster non empty ParStr;
end;

:: PARSP_1:prednot 1 => PARSP_1:pred 1
definition
  let a1 be non empty ParStr;
  let a2, a3, a4, a5 be Element of the carrier of a1;
  pred A2,A3 '||' A4,A5 means
    [a2,a3,a4,a5] in the 4_arg_relation of a1;
end;

:: PARSP_1:dfs 6
definiens
  let a1 be non empty ParStr;
  let a2, a3, a4, a5 be Element of the carrier of a1;
To prove
     a2,a3 '||' a4,a5
it is sufficient to prove
  thus [a2,a3,a4,a5] in the 4_arg_relation of a1;

:: PARSP_1:def 6
theorem
for b1 being non empty ParStr
for b2, b3, b4, b5 being Element of the carrier of b1 holds
   b2,b3 '||' b4,b5
iff
   [b2,b3,b4,b5] in the 4_arg_relation of b1;

:: PARSP_1:funcnot 5 => PARSP_1:func 5
definition
  let a1 be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
  func C3 A1 -> set equals
    [:the carrier of a1,the carrier of a1,the carrier of a1:];
end;

:: PARSP_1:def 7
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr holds
   C3 b1 = [:the carrier of b1,the carrier of b1,the carrier of b1:];

:: PARSP_1:funcreg 1
registration
  let a1 be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
  cluster C3 a1 -> non empty;
end;

:: PARSP_1:funcnot 6 => PARSP_1:func 6
definition
  let a1 be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
  func 4C3 A1 -> set equals
    [:C3 a1,C3 a1,C3 a1,C3 a1:];
end;

:: PARSP_1:def 8
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr holds
   4C3 b1 = [:C3 b1,C3 b1,C3 b1,C3 b1:];

:: PARSP_1:funcreg 2
registration
  let a1 be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
  cluster 4C3 a1 -> non empty;
end;

:: PARSP_1:funcnot 7 => PARSP_1:func 7
definition
  let a1 be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
  func PRs A1 -> set means
    for b1 being set holds
          b1 in it
       iff
          b1 in 4C3 a1 &
           (ex b2, b3, b4, b5 being Element of [:the carrier of a1,the carrier of a1,the carrier of a1:] st
              b1 = [b2,b3,b4,b5] &
               ((b2 `1 - (b3 `1)) * (b4 `2 - (b5 `2))) - ((b4 `1 - (b5 `1)) * (b2 `2 - (b3 `2))) = 0. a1 &
               ((b2 `1 - (b3 `1)) * (b4 `3 - (b5 `3))) - ((b4 `1 - (b5 `1)) * (b2 `3 - (b3 `3))) = 0. a1 &
               ((b2 `2 - (b3 `2)) * (b4 `3 - (b5 `3))) - ((b4 `2 - (b5 `2)) * (b2 `3 - (b3 `3))) = 0. a1);
end;

:: PARSP_1:def 9
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being set holds
      b2 = PRs b1
   iff
      for b3 being set holds
            b3 in b2
         iff
            b3 in 4C3 b1 &
             (ex b4, b5, b6, b7 being Element of [:the carrier of b1,the carrier of b1,the carrier of b1:] st
                b3 = [b4,b5,b6,b7] &
                 ((b4 `1 - (b5 `1)) * (b6 `2 - (b7 `2))) - ((b6 `1 - (b7 `1)) * (b4 `2 - (b5 `2))) = 0. b1 &
                 ((b4 `1 - (b5 `1)) * (b6 `3 - (b7 `3))) - ((b6 `1 - (b7 `1)) * (b4 `3 - (b5 `3))) = 0. b1 &
                 ((b4 `2 - (b5 `2)) * (b6 `3 - (b7 `3))) - ((b6 `2 - (b7 `2)) * (b4 `3 - (b5 `3))) = 0. b1);

:: PARSP_1:th 13
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr holds
   PRs b1 c= [:C3 b1,C3 b1,C3 b1,C3 b1:];

:: PARSP_1:funcnot 8 => PARSP_1:func 8
definition
  let a1 be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
  func PR A1 -> Relation4 of C3 a1 equals
    PRs a1;
end;

:: PARSP_1:def 10
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr holds
   PR b1 = PRs b1;

:: PARSP_1:funcnot 9 => PARSP_1:func 9
definition
  let a1 be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
  func MPS A1 -> ParStr equals
    ParStr(#C3 a1,PR a1#);
end;

:: PARSP_1:def 11
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr holds
   MPS b1 = ParStr(#C3 b1,PR b1#);

:: PARSP_1:funcreg 3
registration
  let a1 be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
  cluster MPS a1 -> non empty strict;
end;

:: PARSP_1:th 16
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr holds
   the carrier of MPS b1 = C3 b1;

:: PARSP_1:th 17
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr holds
   the 4_arg_relation of MPS b1 = PR b1;

:: PARSP_1:th 18
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2, b3, b4, b5 being Element of the carrier of MPS b1 holds
   b2,b3 '||' b4,b5
iff
   [b2,b3,b4,b5] in PR b1;

:: PARSP_1:th 19
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2, b3, b4, b5 being Element of the carrier of MPS b1 holds
   [b2,b3,b4,b5] in PR b1
iff
   [b2,b3,b4,b5] in 4C3 b1 &
    (ex b6, b7, b8, b9 being Element of [:the carrier of b1,the carrier of b1,the carrier of b1:] st
       [b2,b3,b4,b5] = [b6,b7,b8,b9] &
        ((b6 `1 - (b7 `1)) * (b8 `2 - (b9 `2))) - ((b8 `1 - (b9 `1)) * (b6 `2 - (b7 `2))) = 0. b1 &
        ((b6 `1 - (b7 `1)) * (b8 `3 - (b9 `3))) - ((b8 `1 - (b9 `1)) * (b6 `3 - (b7 `3))) = 0. b1 &
        ((b6 `2 - (b7 `2)) * (b8 `3 - (b9 `3))) - ((b8 `2 - (b9 `2)) * (b6 `3 - (b7 `3))) = 0. b1);

:: PARSP_1:th 20
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2, b3, b4, b5 being Element of the carrier of MPS b1 holds
   b2,b3 '||' b4,b5
iff
   [b2,b3,b4,b5] in 4C3 b1 &
    (ex b6, b7, b8, b9 being Element of [:the carrier of b1,the carrier of b1,the carrier of b1:] st
       [b2,b3,b4,b5] = [b6,b7,b8,b9] &
        ((b6 `1 - (b7 `1)) * (b8 `2 - (b9 `2))) - ((b8 `1 - (b9 `1)) * (b6 `2 - (b7 `2))) = 0. b1 &
        ((b6 `1 - (b7 `1)) * (b8 `3 - (b9 `3))) - ((b8 `1 - (b9 `1)) * (b6 `3 - (b7 `3))) = 0. b1 &
        ((b6 `2 - (b7 `2)) * (b8 `3 - (b9 `3))) - ((b8 `2 - (b9 `2)) * (b6 `3 - (b7 `3))) = 0. b1);

:: PARSP_1:th 21
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr holds
   the carrier of MPS b1 = [:the carrier of b1,the carrier of b1,the carrier of b1:];

:: PARSP_1:th 22
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2, b3, b4, b5 being Element of the carrier of MPS b1 holds
[b2,b3,b4,b5] in 4C3 b1;

:: PARSP_1:th 23
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2, b3, b4, b5 being Element of the carrier of MPS b1 holds
   b2,b3 '||' b4,b5
iff
   ex b6, b7, b8, b9 being Element of [:the carrier of b1,the carrier of b1,the carrier of b1:] st
      [b2,b3,b4,b5] = [b6,b7,b8,b9] &
       ((b6 `1 - (b7 `1)) * (b8 `2 - (b9 `2))) - ((b8 `1 - (b9 `1)) * (b6 `2 - (b7 `2))) = 0. b1 &
       ((b6 `1 - (b7 `1)) * (b8 `3 - (b9 `3))) - ((b8 `1 - (b9 `1)) * (b6 `3 - (b7 `3))) = 0. b1 &
       ((b6 `2 - (b7 `2)) * (b8 `3 - (b9 `3))) - ((b8 `2 - (b9 `2)) * (b6 `3 - (b7 `3))) = 0. b1;

:: PARSP_1:th 24
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2, b3 being Element of the carrier of MPS b1 holds
b2,b3 '||' b3,b2;

:: PARSP_1:th 25
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2, b3, b4 being Element of the carrier of MPS b1 holds
b2,b3 '||' b4,b4;

:: PARSP_1:th 26
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2, b3, b4, b5, b6, b7 being Element of the carrier of MPS b1
      st b2,b3 '||' b4,b5 & b2,b3 '||' b6,b7 & not b4,b5 '||' b6,b7
   holds b2 = b3;

:: PARSP_1:th 27
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2, b3, b4 being Element of the carrier of MPS b1
      st b2,b3 '||' b2,b4
   holds b3,b2 '||' b3,b4;

:: PARSP_1:th 28
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2, b3, b4 being Element of the carrier of MPS b1 holds
ex b5 being Element of the carrier of MPS b1 st
   b2,b3 '||' b4,b5 & b2,b4 '||' b3,b5;

:: PARSP_1:attrnot 2 => PARSP_1:attr 2
definition
  let a1 be non empty ParStr;
  attr a1 is ParSp-like means
    for b1, b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of a1 holds
    b1,b2 '||' b2,b1 &
     b1,b2 '||' b3,b3 &
     (b1,b2 '||' b5,b6 & b1,b2 '||' b7,b8 & not b5,b6 '||' b7,b8 implies b1 = b2) &
     (b1,b2 '||' b1,b3 implies b2,b1 '||' b2,b3) &
     (ex b9 being Element of the carrier of a1 st
        b1,b2 '||' b3,b9 & b1,b3 '||' b2,b9);
end;

:: PARSP_1:dfs 12
definiens
  let a1 be non empty ParStr;
To prove
     a1 is ParSp-like
it is sufficient to prove
  thus for b1, b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of a1 holds
    b1,b2 '||' b2,b1 &
     b1,b2 '||' b3,b3 &
     (b1,b2 '||' b5,b6 & b1,b2 '||' b7,b8 & not b5,b6 '||' b7,b8 implies b1 = b2) &
     (b1,b2 '||' b1,b3 implies b2,b1 '||' b2,b3) &
     (ex b9 being Element of the carrier of a1 st
        b1,b2 '||' b3,b9 & b1,b3 '||' b2,b9);

:: PARSP_1:def 12
theorem
for b1 being non empty ParStr holds
      b1 is ParSp-like
   iff
      for b2, b3, b4, b5, b6, b7, b8, b9 being Element of the carrier of b1 holds
      b2,b3 '||' b3,b2 &
       b2,b3 '||' b4,b4 &
       (b2,b3 '||' b6,b7 & b2,b3 '||' b8,b9 & not b6,b7 '||' b8,b9 implies b2 = b3) &
       (b2,b3 '||' b2,b4 implies b3,b2 '||' b3,b4) &
       (ex b10 being Element of the carrier of b1 st
          b2,b3 '||' b4,b10 & b2,b4 '||' b3,b10);

:: PARSP_1:exreg 3
registration
  cluster non empty strict ParSp-like ParStr;
end;

:: PARSP_1:modenot 2
definition
  mode ParSp is non empty ParSp-like ParStr;
end;

:: PARSP_1:th 35
theorem
for b1 being non empty ParSp-like ParStr
for b2, b3 being Element of the carrier of b1 holds
b2,b3 '||' b2,b3;

:: PARSP_1:th 36
theorem
for b1 being non empty ParSp-like ParStr
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2,b3 '||' b4,b5
   holds b4,b5 '||' b2,b3;

:: PARSP_1:th 37
theorem
for b1 being non empty ParSp-like ParStr
for b2, b3, b4 being Element of the carrier of b1 holds
b2,b2 '||' b3,b4;

:: PARSP_1:th 38
theorem
for b1 being non empty ParSp-like ParStr
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2,b3 '||' b4,b5
   holds b3,b2 '||' b4,b5;

:: PARSP_1:th 39
theorem
for b1 being non empty ParSp-like ParStr
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2,b3 '||' b4,b5
   holds b2,b3 '||' b5,b4;

:: PARSP_1:th 40
theorem
for b1 being non empty ParSp-like ParStr
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2,b3 '||' b4,b5
   holds b3,b2 '||' b4,b5 & b2,b3 '||' b5,b4 & b3,b2 '||' b5,b4 & b4,b5 '||' b2,b3 & b5,b4 '||' b2,b3 & b4,b5 '||' b3,b2 & b5,b4 '||' b3,b2;

:: PARSP_1:th 41
theorem
for b1 being non empty ParSp-like ParStr
for b2, b3, b4 being Element of the carrier of b1
      st b2,b3 '||' b2,b4
   holds b2,b4 '||' b2,b3 & b3,b2 '||' b2,b4 & b2,b3 '||' b4,b2 & b2,b4 '||' b3,b2 & b3,b2 '||' b4,b2 & b4,b2 '||' b2,b3 & b4,b2 '||' b3,b2 & b3,b2 '||' b3,b4 & b2,b3 '||' b3,b4 & b3,b2 '||' b4,b3 & b3,b4 '||' b3,b2 & b2,b3 '||' b4,b3 & b4,b3 '||' b3,b2 & b3,b4 '||' b2,b3 & b4,b3 '||' b2,b3 & b4,b2 '||' b4,b3 & b2,b4 '||' b4,b3 & b4,b2 '||' b3,b4 & b2,b4 '||' b3,b4 & b4,b3 '||' b4,b2 & b3,b4 '||' b4,b2 & b4,b3 '||' b2,b4 & b3,b4 '||' b2,b4;

:: PARSP_1:th 42
theorem
for b1 being non empty ParSp-like ParStr
for b2, b3, b4, b5 being Element of the carrier of b1
      st (b2 <> b3 & b4 <> b5 & (b2 = b4 implies b3 <> b5) implies b2 = b5 & b3 = b4)
   holds b2,b3 '||' b4,b5;

:: PARSP_1:th 43
theorem
for b1 being non empty ParSp-like ParStr
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
      st b2 <> b3 & b4,b5 '||' b2,b3 & b2,b3 '||' b6,b7
   holds b4,b5 '||' b6,b7;

:: PARSP_1:th 44
theorem
for b1 being non empty ParSp-like ParStr
for b2, b3, b4 being Element of the carrier of b1
      st not b2,b3 '||' b2,b4
   holds b2 <> b3 & b3 <> b4 & b4 <> b2;

:: PARSP_1:th 45
theorem
for b1 being non empty ParSp-like ParStr
for b2, b3, b4, b5 being Element of the carrier of b1
      st not b2,b3 '||' b4,b5
   holds b2 <> b3 & b4 <> b5;

:: PARSP_1:th 47
theorem
for b1 being non empty ParSp-like ParStr
for b2, b3, b4 being Element of the carrier of b1
      st not b2,b3 '||' b2,b4
   holds not b2,b4 '||' b2,b3 & not b3,b2 '||' b2,b4 & not b2,b3 '||' b4,b2 & not b2,b4 '||' b3,b2 & not b3,b2 '||' b4,b2 & not b4,b2 '||' b2,b3 & not b4,b2 '||' b3,b2 & not b3,b2 '||' b3,b4 & not b2,b3 '||' b3,b4 & not b3,b2 '||' b4,b3 & not b3,b4 '||' b3,b2 & not b3,b2 '||' b4,b3 & not b4,b3 '||' b3,b2 & not b3,b4 '||' b2,b3 & not b4,b3 '||' b2,b3 & not b4,b2 '||' b4,b3 & not b2,b4 '||' b4,b3 & not b4,b2 '||' b3,b4 & not b2,b4 '||' b3,b4 & not b4,b3 '||' b4,b2 & not b3,b4 '||' b4,b2 & not b4,b3 '||' b2,b4 & not b3,b4 '||' b2,b4;

:: PARSP_1:th 48
theorem
for b1 being non empty ParSp-like ParStr
for b2, b3, b4, b5, b6, b7, b8, b9 being Element of the carrier of b1
      st not b2,b3 '||' b4,b5 & b2,b3 '||' b6,b7 & b4,b5 '||' b8,b9 & b6 <> b7 & b8 <> b9
   holds not b6,b7 '||' b8,b9;

:: PARSP_1:th 49
theorem
for b1 being non empty ParSp-like ParStr
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
      st not b2,b3 '||' b2,b4 & b2,b3 '||' b5,b6 & b2,b4 '||' b5,b7 & b3,b4 '||' b6,b7 & b5 <> b6
   holds not b5,b6 '||' b5,b7;

:: PARSP_1:th 50
theorem
for b1 being non empty ParSp-like ParStr
for b2, b3, b4, b5, b6 being Element of the carrier of b1
      st not b2,b3 '||' b2,b4 & b2,b4 '||' b5,b6 & b3,b4 '||' b5,b6
   holds b5 = b6;

:: PARSP_1:th 51
theorem
for b1 being non empty ParSp-like ParStr
for b2, b3, b4, b5 being Element of the carrier of b1
      st not b2,b3 '||' b2,b4 & b2,b4 '||' b2,b5 & b3,b4 '||' b3,b5
   holds b4 = b5;

:: PARSP_1:th 52
theorem
for b1 being non empty ParSp-like ParStr
for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of b1
      st not b2,b3 '||' b2,b4 & b2,b3 '||' b5,b6 & b2,b4 '||' b5,b7 & b2,b4 '||' b5,b8 & b3,b4 '||' b6,b7 & b3,b4 '||' b6,b8
   holds b7 = b8;

:: PARSP_1:th 53
theorem
for b1 being non empty ParSp-like ParStr
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2,b3 '||' b2,b4 & b2,b3 '||' b2,b5
   holds b2,b3 '||' b4,b5;

:: PARSP_1:th 54
theorem
for b1 being non empty ParSp-like ParStr
   st for b2, b3 being Element of the carrier of b1 holds
     b2 = b3
for b2, b3, b4, b5 being Element of the carrier of b1 holds
b2,b3 '||' b4,b5;

:: PARSP_1:th 55
theorem
for b1 being non empty ParSp-like ParStr
   st ex b2, b3 being Element of the carrier of b1 st
        b2 <> b3 &
         (for b4 being Element of the carrier of b1 holds
            b2,b3 '||' b2,b4)
for b2, b3, b4, b5 being Element of the carrier of b1 holds
b2,b3 '||' b4,b5;

:: PARSP_1:th 56
theorem
for b1 being non empty ParSp-like ParStr
for b2, b3, b4, b5, b6 being Element of the carrier of b1
      st not b2,b3 '||' b2,b4 & b5 <> b6 & b5,b6 '||' b5,b2 & b5,b6 '||' b5,b3
   holds not b5,b6 '||' b5,b4;