Article VECTSP_4, MML version 4.99.1005

:: VECTSP_4:attrnot 1 => VECTSP_4:attr 1
definition
  let a1 be non empty multMagma;
  let a2 be non empty VectSpStr over a1;
  let a3 be Element of bool the carrier of a2;
  attr a3 is linearly-closed means
    (for b1, b2 being Element of the carrier of a2
           st b1 in a3 & b2 in a3
        holds b1 + b2 in a3) &
     (for b1 being Element of the carrier of a1
     for b2 being Element of the carrier of a2
           st b2 in a3
        holds b1 * b2 in a3);
end;

:: VECTSP_4:dfs 1
definiens
  let a1 be non empty multMagma;
  let a2 be non empty VectSpStr over a1;
  let a3 be Element of bool the carrier of a2;
To prove
     a3 is linearly-closed
it is sufficient to prove
  thus (for b1, b2 being Element of the carrier of a2
           st b1 in a3 & b2 in a3
        holds b1 + b2 in a3) &
     (for b1 being Element of the carrier of a1
     for b2 being Element of the carrier of a2
           st b2 in a3
        holds b1 * b2 in a3);

:: VECTSP_4:def 1
theorem
for b1 being non empty multMagma
for b2 being non empty VectSpStr over b1
for b3 being Element of bool the carrier of b2 holds
      b3 is linearly-closed(b1, b2)
   iff
      (for b4, b5 being Element of the carrier of b2
             st b4 in b3 & b5 in b3
          holds b4 + b5 in b3) &
       (for b4 being Element of the carrier of b1
       for b5 being Element of the carrier of b2
             st b5 in b3
          holds b4 * b5 in b3);

:: VECTSP_4:th 4
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of bool the carrier of b2
      st b3 <> {} & b3 is linearly-closed(b1, b2)
   holds 0. b2 in b3;

:: VECTSP_4:th 5
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of bool the carrier of b2
   st b3 is linearly-closed(b1, b2)
for b4 being Element of the carrier of b2
      st b4 in b3
   holds - b4 in b3;

:: VECTSP_4:th 6
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of bool the carrier of b2
   st b3 is linearly-closed(b1, b2)
for b4, b5 being Element of the carrier of b2
      st b4 in b3 & b5 in b3
   holds b4 - b5 in b3;

:: VECTSP_4:th 7
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1 holds
   {0. b2} is linearly-closed(b1, b2);

:: VECTSP_4:th 8
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of bool the carrier of b2
      st the carrier of b2 = b3
   holds b3 is linearly-closed(b1, b2);

:: VECTSP_4:th 9
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4, b5 being Element of bool the carrier of b2
      st b3 is linearly-closed(b1, b2) &
         b4 is linearly-closed(b1, b2) &
         b5 = {b6 + b7 where b6 is Element of the carrier of b2, b7 is Element of the carrier of b2: b6 in b3 & b7 in b4}
   holds b5 is linearly-closed(b1, b2);

:: VECTSP_4:th 10
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Element of bool the carrier of b2
      st b3 is linearly-closed(b1, b2) & b4 is linearly-closed(b1, b2)
   holds b3 /\ b4 is linearly-closed(b1, b2);

:: VECTSP_4:modenot 1 => VECTSP_4:mode 1
definition
  let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
  let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
  mode Subspace of A2 -> non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1 means
    the carrier of it c= the carrier of a2 &
     0. it = 0. a2 &
     the addF of it = (the addF of a2) || the carrier of it &
     the lmult of it = (the lmult of a2) | [:the carrier of a1,the carrier of it:];
end;

:: VECTSP_4:dfs 2
definiens
  let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
  let a2, a3 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
To prove
     a3 is Subspace of a2
it is sufficient to prove
  thus the carrier of a3 c= the carrier of a2 &
     0. a3 = 0. a2 &
     the addF of a3 = (the addF of a2) || the carrier of a3 &
     the lmult of a3 = (the lmult of a2) | [:the carrier of a1,the carrier of a3:];

:: VECTSP_4:def 2
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2, b3 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1 holds
   b3 is Subspace of b2
iff
   the carrier of b3 c= the carrier of b2 &
    0. b3 = 0. b2 &
    the addF of b3 = (the addF of b2) || the carrier of b3 &
    the lmult of b3 = (the lmult of b2) | [:the carrier of b1,the carrier of b3:];

:: VECTSP_4:th 16
theorem
for b1 being set
for b2 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b3 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b2
for b4, b5 being Subspace of b3
      st b1 in b4 & b4 is Subspace of b5
   holds b1 in b5;

:: VECTSP_4:th 17
theorem
for b1 being set
for b2 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b3 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b2
for b4 being Subspace of b3
      st b1 in b4
   holds b1 in b3;

:: VECTSP_4:th 18
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Subspace of b2
for b4 being Element of the carrier of b3 holds
   b4 is Element of the carrier of b2;

:: VECTSP_4:th 19
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Subspace of b2 holds
   0. b3 = 0. b2;

:: VECTSP_4:th 20
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Subspace of b2 holds
0. b3 = 0. b4;

:: VECTSP_4:th 21
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Element of the carrier of b2
for b5 being Subspace of b2
for b6, b7 being Element of the carrier of b5
      st b6 = b3 & b7 = b4
   holds b6 + b7 = b3 + b4;

:: VECTSP_4:th 22
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of the carrier of b1
for b4 being Element of the carrier of b2
for b5 being Subspace of b2
for b6 being Element of the carrier of b5
      st b6 = b4
   holds b3 * b6 = b3 * b4;

:: VECTSP_4:th 23
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of the carrier of b2
for b4 being Subspace of b2
for b5 being Element of the carrier of b4
      st b5 = b3
   holds - b3 = - b5;

:: VECTSP_4:th 24
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Element of the carrier of b2
for b5 being Subspace of b2
for b6, b7 being Element of the carrier of b5
      st b6 = b3 & b7 = b4
   holds b6 - b7 = b3 - b4;

:: VECTSP_4:th 25
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Subspace of b2 holds
   0. b2 in b3;

:: VECTSP_4:th 26
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Subspace of b2 holds
0. b3 in b4;

:: VECTSP_4:th 27
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Subspace of b2 holds
   0. b3 in b2;

:: VECTSP_4:th 28
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Element of the carrier of b2
for b5 being Subspace of b2
      st b3 in b5 & b4 in b5
   holds b3 + b4 in b5;

:: VECTSP_4:th 29
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of the carrier of b1
for b4 being Element of the carrier of b2
for b5 being Subspace of b2
      st b4 in b5
   holds b3 * b4 in b5;

:: VECTSP_4:th 30
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of the carrier of b2
for b4 being Subspace of b2
      st b3 in b4
   holds - b3 in b4;

:: VECTSP_4:th 31
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Element of the carrier of b2
for b5 being Subspace of b2
      st b3 in b5 & b4 in b5
   holds b3 - b4 in b5;

:: VECTSP_4:th 32
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1 holds
   b2 is Subspace of b2;

:: VECTSP_4:th 33
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2, b3 being non empty right_complementable strict VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
      st b3 is Subspace of b2 & b2 is Subspace of b3
   holds b3 = b2;

:: VECTSP_4:th 34
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2, b3, b4 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
      st b2 is Subspace of b3 & b3 is Subspace of b4
   holds b2 is Subspace of b4;

:: VECTSP_4:th 35
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Subspace of b2
      st the carrier of b3 c= the carrier of b4
   holds b3 is Subspace of b4;

:: VECTSP_4:th 36
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Subspace of b2
      st for b5 being Element of the carrier of b2
              st b5 in b3
           holds b5 in b4
   holds b3 is Subspace of b4;

:: VECTSP_4:exreg 1
registration
  let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
  let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
  cluster non empty right_complementable strict VectSp-like Abelian add-associative right_zeroed Subspace of a2;
end;

:: VECTSP_4:th 37
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being strict Subspace of b2
      st the carrier of b3 = the carrier of b4
   holds b3 = b4;

:: VECTSP_4:th 38
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being strict Subspace of b2
      st for b5 being Element of the carrier of b2 holds
              b5 in b3
           iff
              b5 in b4
   holds b3 = b4;

:: VECTSP_4:th 39
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable strict VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being strict Subspace of b2
      st the carrier of b3 = the carrier of b2
   holds b3 = b2;

:: VECTSP_4:th 40
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable strict VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being strict Subspace of b2
      st for b4 being Element of the carrier of b2 holds
           b4 in b3
   holds b3 = b2;

:: VECTSP_4:th 41
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Subspace of b2
for b4 being Element of bool the carrier of b2
      st the carrier of b3 = b4
   holds b4 is linearly-closed(b1, b2);

:: VECTSP_4:th 42
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of bool the carrier of b2
      st b3 <> {} & b3 is linearly-closed(b1, b2)
   holds ex b4 being strict Subspace of b2 st
      b3 = the carrier of b4;

:: VECTSP_4:funcnot 1 => VECTSP_4:func 1
definition
  let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
  let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
  func (0). A2 -> strict Subspace of a2 means
    the carrier of it = {0. a2};
end;

:: VECTSP_4:def 3
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being strict Subspace of b2 holds
      b3 = (0). b2
   iff
      the carrier of b3 = {0. b2};

:: VECTSP_4:funcnot 2 => VECTSP_4:func 2
definition
  let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
  let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
  func (Omega). A2 -> strict Subspace of a2 equals
    VectSpStr(#the carrier of a2,the addF of a2,the ZeroF of a2,the lmult of a2#);
end;

:: VECTSP_4:def 4
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1 holds
   (Omega). b2 = VectSpStr(#the carrier of b2,the addF of b2,the ZeroF of b2,the lmult of b2#);

:: VECTSP_4:th 46
theorem
for b1 being set
for b2 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b3 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b2 holds
      b1 in (0). b3
   iff
      b1 = 0. b3;

:: VECTSP_4:th 47
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Subspace of b2 holds
   (0). b3 = (0). b2;

:: VECTSP_4:th 48
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Subspace of b2 holds
(0). b3 = (0). b4;

:: VECTSP_4:th 49
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Subspace of b2 holds
   (0). b3 is Subspace of b2;

:: VECTSP_4:th 50
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Subspace of b2 holds
   (0). b2 is Subspace of b3;

:: VECTSP_4:th 51
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Subspace of b2 holds
(0). b3 is Subspace of b4;

:: VECTSP_4:th 53
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable strict VectSp-like Abelian add-associative right_zeroed VectSpStr over b1 holds
   b2 is Subspace of (Omega). b2;

:: VECTSP_4:funcnot 3 => VECTSP_4:func 3
definition
  let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
  let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
  let a3 be Element of the carrier of a2;
  let a4 be Subspace of a2;
  func A3 + A4 -> Element of bool the carrier of a2 equals
    {a3 + b1 where b1 is Element of the carrier of a2: b1 in a4};
end;

:: VECTSP_4:def 5
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of the carrier of b2
for b4 being Subspace of b2 holds
   b3 + b4 = {b3 + b5 where b5 is Element of the carrier of b2: b5 in b4};

:: VECTSP_4:modenot 2 => VECTSP_4:mode 2
definition
  let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
  let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
  let a3 be Subspace of a2;
  mode Coset of A3 -> Element of bool the carrier of a2 means
    ex b1 being Element of the carrier of a2 st
       it = b1 + a3;
end;

:: VECTSP_4:dfs 6
definiens
  let a1 be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
  let a2 be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over a1;
  let a3 be Subspace of a2;
  let a4 be Element of bool the carrier of a2;
To prove
     a4 is Coset of a3
it is sufficient to prove
  thus ex b1 being Element of the carrier of a2 st
       a4 = b1 + a3;

:: VECTSP_4:def 6
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Subspace of b2
for b4 being Element of bool the carrier of b2 holds
      b4 is Coset of b3
   iff
      ex b5 being Element of the carrier of b2 st
         b4 = b5 + b3;

:: VECTSP_4:th 57
theorem
for b1 being set
for b2 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b3 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b2
for b4 being Element of the carrier of b3
for b5 being Subspace of b3 holds
      b1 in b4 + b5
   iff
      ex b6 being Element of the carrier of b3 st
         b6 in b5 & b1 = b4 + b6;

:: VECTSP_4:th 58
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of the carrier of b2
for b4 being Subspace of b2 holds
      0. b2 in b3 + b4
   iff
      b3 in b4;

:: VECTSP_4:th 59
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of the carrier of b2
for b4 being Subspace of b2 holds
   b3 in b3 + b4;

:: VECTSP_4:th 60
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Subspace of b2 holds
   (0. b2) + b3 = the carrier of b3;

:: VECTSP_4:th 61
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of the carrier of b2 holds
   b3 + (0). b2 = {b3};

:: VECTSP_4:th 62
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of the carrier of b2 holds
   b3 + (Omega). b2 = the carrier of b2;

:: VECTSP_4:th 63
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of the carrier of b2
for b4 being Subspace of b2 holds
      0. b2 in b3 + b4
   iff
      b3 + b4 = the carrier of b4;

:: VECTSP_4:th 64
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of the carrier of b2
for b4 being Subspace of b2 holds
      b3 in b4
   iff
      b3 + b4 = the carrier of b4;

:: VECTSP_4:th 65
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of the carrier of b1
for b4 being Element of the carrier of b2
for b5 being Subspace of b2
      st b4 in b5
   holds (b3 * b4) + b5 = the carrier of b5;

:: VECTSP_4:th 66
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 non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of the carrier of b1
for b4 being Element of the carrier of b2
for b5 being Subspace of b2
      st b3 <> 0. b1 & (b3 * b4) + b5 = the carrier of b5
   holds b4 in b5;

:: VECTSP_4:th 67
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 non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of the carrier of b2
for b4 being Subspace of b2 holds
      b3 in b4
   iff
      (- b3) + b4 = the carrier of b4;

:: VECTSP_4:th 68
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Element of the carrier of b2
for b5 being Subspace of b2 holds
      b3 in b5
   iff
      b4 + b5 = (b4 + b3) + b5;

:: VECTSP_4:th 69
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Element of the carrier of b2
for b5 being Subspace of b2 holds
      b3 in b5
   iff
      b4 + b5 = (b4 - b3) + b5;

:: VECTSP_4:th 70
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Element of the carrier of b2
for b5 being Subspace of b2 holds
      b3 in b4 + b5
   iff
      b4 + b5 = b3 + b5;

:: VECTSP_4:th 71
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4, b5 being Element of the carrier of b2
for b6 being Subspace of b2
      st b3 in b4 + b6 & b3 in b5 + b6
   holds b4 + b6 = b5 + b6;

:: VECTSP_4:th 72
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 non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of the carrier of b1
for b4 being Element of the carrier of b2
for b5 being Subspace of b2
      st b3 <> 1_ b1 & b3 * b4 in b4 + b5
   holds b4 in b5;

:: VECTSP_4:th 73
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of the carrier of b1
for b4 being Element of the carrier of b2
for b5 being Subspace of b2
      st b4 in b5
   holds b3 * b4 in b4 + b5;

:: VECTSP_4:th 74
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of the carrier of b2
for b4 being Subspace of b2
      st b3 in b4
   holds - b3 in b3 + b4;

:: VECTSP_4:th 75
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Element of the carrier of b2
for b5 being Subspace of b2 holds
      b3 + b4 in b4 + b5
   iff
      b3 in b5;

:: VECTSP_4:th 76
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Element of the carrier of b2
for b5 being Subspace of b2 holds
      b3 - b4 in b3 + b5
   iff
      b4 in b5;

:: VECTSP_4:th 78
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Element of the carrier of b2
for b5 being Subspace of b2 holds
      b3 in b4 + b5
   iff
      ex b6 being Element of the carrier of b2 st
         b6 in b5 & b3 = b4 - b6;

:: VECTSP_4:th 79
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Element of the carrier of b2
for b5 being Subspace of b2 holds
      ex b6 being Element of the carrier of b2 st
         b3 in b6 + b5 & b4 in b6 + b5
   iff
      b3 - b4 in b5;

:: VECTSP_4:th 80
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Element of the carrier of b2
for b5 being Subspace of b2
      st b3 + b5 = b4 + b5
   holds ex b6 being Element of the carrier of b2 st
      b6 in b5 & b3 + b6 = b4;

:: VECTSP_4:th 81
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Element of the carrier of b2
for b5 being Subspace of b2
      st b3 + b5 = b4 + b5
   holds ex b6 being Element of the carrier of b2 st
      b6 in b5 & b3 - b6 = b4;

:: VECTSP_4:th 82
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of the carrier of b2
for b4, b5 being strict Subspace of b2 holds
   b3 + b4 = b3 + b5
iff
   b4 = b5;

:: VECTSP_4:th 83
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Element of the carrier of b2
for b5, b6 being strict Subspace of b2
      st b3 + b5 = b4 + b6
   holds b5 = b6;

:: VECTSP_4:th 84
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of the carrier of b2
for b4 being Subspace of b2 holds
   ex b5 being Coset of b4 st
      b3 in b5;

:: VECTSP_4:th 85
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Subspace of b2
for b4 being Coset of b3 holds
      b4 is linearly-closed(b1, b2)
   iff
      b4 = the carrier of b3;

:: VECTSP_4:th 86
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being strict Subspace of b2
for b5 being Coset of b3
for b6 being Coset of b4
      st b5 = b6
   holds b3 = b4;

:: VECTSP_4:th 87
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of the carrier of b2 holds
   {b3} is Coset of (0). b2;

:: VECTSP_4:th 88
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of bool the carrier of b2
      st b3 is Coset of (0). b2
   holds ex b4 being Element of the carrier of b2 st
      b3 = {b4};

:: VECTSP_4:th 89
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Subspace of b2 holds
   the carrier of b3 is Coset of b3;

:: VECTSP_4:th 90
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1 holds
   the carrier of b2 is Coset of (Omega). b2;

:: VECTSP_4:th 91
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of bool the carrier of b2
      st b3 is Coset of (Omega). b2
   holds b3 = the carrier of b2;

:: VECTSP_4:th 92
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Subspace of b2
for b4 being Coset of b3 holds
      0. b2 in b4
   iff
      b4 = the carrier of b3;

:: VECTSP_4:th 93
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of the carrier of b2
for b4 being Subspace of b2
for b5 being Coset of b4 holds
      b3 in b5
   iff
      b5 = b3 + b4;

:: VECTSP_4:th 94
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Element of the carrier of b2
for b5 being Subspace of b2
for b6 being Coset of b5
      st b3 in b6 & b4 in b6
   holds ex b7 being Element of the carrier of b2 st
      b7 in b5 & b3 + b7 = b4;

:: VECTSP_4:th 95
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Element of the carrier of b2
for b5 being Subspace of b2
for b6 being Coset of b5
      st b3 in b6 & b4 in b6
   holds ex b7 being Element of the carrier of b2 st
      b7 in b5 & b3 - b7 = b4;

:: VECTSP_4:th 96
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Element of the carrier of b2
for b5 being Subspace of b2 holds
      ex b6 being Coset of b5 st
         b3 in b6 & b4 in b6
   iff
      b3 - b4 in b5;

:: VECTSP_4:th 97
theorem
for b1 being non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3 being Element of the carrier of b2
for b4 being Subspace of b2
for b5, b6 being Coset of b4
      st b3 in b5 & b3 in b6
   holds b5 = b6;

:: VECTSP_4:th 103
theorem
for b1 being non empty right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr over b1
for b3, b4 being Element of the carrier of b1
for b5 being Element of the carrier of b2 holds
   (b3 - b4) * b5 = (b3 * b5) - (b4 * b5);