Article RANKNULL, MML version 4.99.1005

:: RANKNULL:th 1
theorem
for b1, b2 being Relation-like Function-like set
      st b2 is one-to-one & b1 | proj2 b2 is one-to-one & proj2 b2 c= proj1 b1
   holds b2 * b1 is one-to-one;

:: RANKNULL:th 2
theorem
for b1 being Relation-like Function-like set
for b2, b3 being set
      st b2 c= b3 & b1 | b3 is one-to-one
   holds b1 | b2 is one-to-one;

:: RANKNULL:th 3
theorem
for b1 being 1-sorted
for b2, b3 being Element of bool the carrier of b1 holds
   b2 meets b3
iff
   ex b4 being Element of the carrier of b1 st
      b4 in b2 & b4 in b3;

:: RANKNULL:exreg 1
registration
  let a1 be non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like finite-dimensional VectSpStr over a1;
  cluster finite Basis of a2;
end;

:: RANKNULL:exreg 2
registration
  let a1 be non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
  let a2, a3 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over a1;
  cluster Relation-like Function-like non empty quasi_total linear total Relation of the carrier of a2,the carrier of a3;
end;

:: RANKNULL:th 4
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 non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
      st [#] b2 is finite
   holds b2 is finite-dimensional(b1);

:: RANKNULL:th 5
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 non empty right_complementable Abelian add-associative right_zeroed VectSp-like finite-dimensional VectSpStr over b1
      st Card [#] b2 = 1
   holds dim b2 = {};

:: RANKNULL: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 non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
      st Card [#] b2 = 2
   holds dim b2 = 1;

:: RANKNULL:modenot 1
definition
  let a1 be non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
  let a2, a3 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over a1;
  mode linear-transformation of a2,a3 is Function-like quasi_total linear Relation of the carrier of a2,the carrier of a3;
end;

:: RANKNULL:th 7
theorem
for b1, b2 being non empty 1-sorted
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
   dom b3 = [#] b1 & proj2 b3 c= [#] b2;

:: RANKNULL:th 8
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 non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total linear Relation of the carrier of b3,the carrier of b2
for b5, b6 being Element of the carrier of b3 holds
(b4 . b5) - (b4 . b6) = b4 . (b5 - b6);

:: RANKNULL:th 9
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 non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total linear Relation of the carrier of b2,the carrier of b3 holds
   b4 . 0. b2 = 0. b3;

:: RANKNULL:funcnot 1 => RANKNULL:func 1
definition
  let a1 be non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
  let a2, a3 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over a1;
  let a4 be Function-like quasi_total linear Relation of the carrier of a2,the carrier of a3;
  func ker A4 -> strict Subspace of a2 means
    [#] it = {b1 where b1 is Element of the carrier of a2: a4 . b1 = 0. a3};
end;

:: RANKNULL:def 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
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total linear Relation of the carrier of b2,the carrier of b3
for b5 being strict Subspace of b2 holds
      b5 = ker b4
   iff
      [#] b5 = {b6 where b6 is Element of the carrier of b2: b4 . b6 = 0. b3};

:: RANKNULL:th 10
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 non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total linear Relation of the carrier of b2,the carrier of b3
for b5 being Element of the carrier of b2 holds
      b5 in ker b4
   iff
      b4 . b5 = 0. b3;

:: RANKNULL:funcnot 2 => RANKNULL:func 2
definition
  let a1, a2 be non empty 1-sorted;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
  let a4 be Element of bool the carrier of a1;
  redefine func a3 .: a4 -> Element of bool the carrier of a2;
end;

:: RANKNULL:funcnot 3 => RANKNULL:func 3
definition
  let a1 be non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
  let a2, a3 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over a1;
  let a4 be Function-like quasi_total linear Relation of the carrier of a2,the carrier of a3;
  func im A4 -> strict Subspace of a3 means
    [#] it = a4 .: [#] a2;
end;

:: RANKNULL:def 2
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 non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total linear Relation of the carrier of b2,the carrier of b3
for b5 being strict Subspace of b3 holds
      b5 = im b4
   iff
      [#] b5 = b4 .: [#] b2;

:: RANKNULL:th 11
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 non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total linear Relation of the carrier of b3,the carrier of b2 holds
   0. b3 in ker b4;

:: RANKNULL: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 non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total linear Relation of the carrier of b3,the carrier of b2
for b5 being Element of bool the carrier of b3 holds
   b4 .: b5 is Element of bool the carrier of im b4;

:: RANKNULL: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, b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total linear Relation of the carrier of b3,the carrier of b2
for b5 being Element of the carrier of b2 holds
      b5 in im b4
   iff
      ex b6 being Element of the carrier of b3 st
         b5 = b4 . b6;

:: RANKNULL: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 non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total linear Relation of the carrier of b2,the carrier of b3
for b5 being Element of the carrier of ker b4 holds
   b4 . b5 = 0. b3;

:: RANKNULL: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 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total linear Relation of the carrier of b3,the carrier of b2
      st b4 is one-to-one
   holds ker b4 = (0). b3;

:: RANKNULL: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 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like finite-dimensional VectSpStr over b1 holds
   dim (0). b2 = {};

:: RANKNULL: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, b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total linear Relation of the carrier of b3,the carrier of b2
for b5, b6 being Element of the carrier of b3
      st b4 . b5 = b4 . b6
   holds b5 - b6 in ker b4;

:: RANKNULL: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 non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3 being Element of bool the carrier of b2
for b4, b5 being Element of the carrier of b2
      st b4 - b5 in Lin b3
   holds b4 in Lin (b3 \/ {b5});

:: RANKNULL: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 non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Element of bool the carrier of b2
      st b2 is Subspace of b3
   holds b4 is Element of bool the carrier of b3;

:: RANKNULL: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
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3 being Element of bool the carrier of b2
      st b3 is linearly-independent(b1, b2)
   holds b3 is Basis of Lin b3;

:: RANKNULL: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
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3 being Element of bool the carrier of b2
for b4 being Element of the carrier of b2
      st b4 in Lin b3 & not b4 in b3
   holds b3 \/ {b4} is linearly-dependent(b1, b2);

:: RANKNULL: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 non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total linear Relation of the carrier of b3,the carrier of b2
for b5 being Element of bool the carrier of b3
for b6 being Basis of b3
      st b5 is Basis of ker b4 & b5 c= b6
   holds b4 | (b6 \ b5) is one-to-one;

:: RANKNULL:th 23
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 non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3 being Element of bool the carrier of b2
for b4 being Linear_Combination of b3
for b5 being Element of the carrier of b2
for b6 being Element of the carrier of b1 holds
   b4 +*(b5,b6) is Linear_Combination of b3 \/ {b5};

:: RANKNULL:funcnot 4 => RANKNULL:func 4
definition
  let a1 be 1-sorted;
  let a2 be Element of bool the carrier of a1;
  func A1 \ A2 -> Element of bool the carrier of a1 equals
    ([#] a1) \ a2;
end;

:: RANKNULL:def 3
theorem
for b1 being 1-sorted
for b2 being Element of bool the carrier of b1 holds
   b1 \ b2 = ([#] b1) \ b2;

:: RANKNULL:funcnot 5 => RANKNULL: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;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over a1;
  let a3 be Linear_Combination of a2;
  let a4 be Element of bool the carrier of a2;
  redefine func a3 .: a4 -> Element of bool the carrier of a1;
end;

:: RANKNULL:exreg 3
registration
  let a1 be non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over a1;
  cluster linearly-dependent Element of bool the carrier of a2;
end;

:: RANKNULL:funcnot 6 => RANKNULL:func 6
definition
  let a1 be non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over a1;
  let a3 be Linear_Combination of a2;
  let a4 be Element of bool the carrier of a2;
  func A3 ! A4 -> Linear_Combination of a4 equals
    (a3 | a4) +* ((a2 \ a4) --> 0. a1);
end;

:: RANKNULL:def 4
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 non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3 being Linear_Combination of b2
for b4 being Element of bool the carrier of b2 holds
   b3 ! b4 = (b3 | b4) +* ((b2 \ b4) --> 0. b1);

:: RANKNULL:th 24
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 non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3 being Linear_Combination of b2 holds
   b3 = b3 ! Carrier b3;

:: RANKNULL:th 25
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 non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3 being Linear_Combination of b2
for b4 being Element of bool the carrier of b2
for b5 being Element of the carrier of b2
      st b5 in b4
   holds (b3 ! b4) . b5 = b3 . b5;

:: RANKNULL: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 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3 being Linear_Combination of b2
for b4 being Element of bool the carrier of b2
for b5 being Element of the carrier of b2
      st not b5 in b4
   holds (b3 ! b4) . b5 = 0. b1;

:: RANKNULL: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 non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3, b4 being Element of bool the carrier of b2
for b5 being Linear_Combination of b4
      st b3 c= b4
   holds b5 = (b5 ! b3) + (b5 ! (b4 \ b3));

:: RANKNULL:funcreg 1
registration
  let a1 be non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over a1;
  let a3 be Linear_Combination of a2;
  let a4 be Element of bool the carrier of a2;
  cluster a3 .: a4 -> finite;
end;

:: RANKNULL:funcnot 7 => RANKNULL:func 7
definition
  let a1, a2 be non empty 1-sorted;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
  let a4 be Element of bool the carrier of a2;
  redefine func a3 " a4 -> Element of bool the carrier of a1;
end;

:: RANKNULL: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 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3 being Linear_Combination of b2
for b4 being Element of bool the carrier of b2
      st b4 misses Carrier b3
   holds b3 .: b4 c= {0. b1};

:: RANKNULL:funcnot 8 => RANKNULL:func 8
definition
  let a1 be non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
  let a2, a3 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over a1;
  let a4 be Linear_Combination of a2;
  let a5 be Function-like quasi_total linear Relation of the carrier of a2,the carrier of a3;
  func A5 @ A4 -> Linear_Combination of a3 means
    for b1 being Element of the carrier of a3 holds
       it . b1 = Sum (a4 .: (a5 " {b1}));
end;

:: RANKNULL:def 5
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 non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Linear_Combination of b2
for b5 being Function-like quasi_total linear Relation of the carrier of b2,the carrier of b3
for b6 being Linear_Combination of b3 holds
      b6 = b5 @ b4
   iff
      for b7 being Element of the carrier of b3 holds
         b6 . b7 = Sum (b4 .: (b5 " {b7}));

:: RANKNULL: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 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total linear Relation of the carrier of b3,the carrier of b2
for b5 being Linear_Combination of b3 holds
   b4 @ b5 is Linear_Combination of b4 .: Carrier b5;

:: RANKNULL: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, b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total linear Relation of the carrier of b3,the carrier of b2
for b5 being Linear_Combination of b3 holds
   Carrier (b4 @ b5) c= b4 .: Carrier b5;

:: RANKNULL:th 31
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 non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3, b4 being Linear_Combination of b2
      st Carrier b3 misses Carrier b4
   holds Carrier (b3 + b4) = (Carrier b3) \/ Carrier b4;

:: RANKNULL:th 32
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 non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3, b4 being Linear_Combination of b2
      st Carrier b3 misses Carrier b4
   holds Carrier (b3 - b4) = (Carrier b3) \/ Carrier b4;

:: RANKNULL:th 33
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 non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3, b4 being Element of bool the carrier of b2
      st b3 c= b4 & b4 is Basis of b2
   holds b2 is_the_direct_sum_of Lin b3,Lin (b4 \ b3);

:: RANKNULL:th 34
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 non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total linear Relation of the carrier of b3,the carrier of b2
for b5 being Element of bool the carrier of b3
for b6 being Linear_Combination of b5
for b7 being Element of the carrier of b3
      st b4 | b5 is one-to-one & b7 in b5
   holds ex b8 being Element of bool the carrier of b3 st
      b8 misses b5 &
       b4 " {b4 . b7} = {b7} \/ b8;

:: RANKNULL:th 35
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 non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3 being Linear_Combination of b2
for b4 being Element of bool the carrier of b2
      st b4 misses Carrier b3 & b4 <> {}
   holds b3 .: b4 = {0. b1};

:: RANKNULL:th 36
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 non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total linear Relation of the carrier of b2,the carrier of b3
for b5 being Linear_Combination of b2
for b6 being Element of the carrier of b3
      st b6 in Carrier (b4 @ b5)
   holds b4 " {b6} meets Carrier b5;

:: RANKNULL:th 37
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 non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total linear Relation of the carrier of b3,the carrier of b2
for b5 being Linear_Combination of b3
for b6 being Element of the carrier of b3
      st b4 | Carrier b5 is one-to-one & b6 in Carrier b5
   holds (b4 @ b5) . (b4 . b6) = b5 . b6;

:: RANKNULL:th 38
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 non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total linear Relation of the carrier of b3,the carrier of b2
for b5 being Linear_Combination of b3
for b6 being FinSequence of the carrier of b3
      st proj2 b6 = Carrier b5 & b4 | Carrier b5 is one-to-one
   holds b4 * (b5 (#) b6) = (b4 @ b5) (#) (b4 * b6);

:: RANKNULL:th 39
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 non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total linear Relation of the carrier of b3,the carrier of b2
for b5 being Linear_Combination of b3
      st b4 | Carrier b5 is one-to-one
   holds b4 .: Carrier b5 = Carrier (b4 @ b5);

:: RANKNULL:th 40
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 non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total linear Relation of the carrier of b3,the carrier of b2
for b5 being Element of bool the carrier of b3
for b6 being Basis of b3
for b7 being Linear_Combination of b6 \ b5
      st b5 is Basis of ker b4 & b5 c= b6
   holds b4 . Sum b7 = Sum (b4 @ b7);

:: RANKNULL:th 41
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 non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3 being Element of bool the carrier of b2
      st b3 is linearly-dependent(b1, b2)
   holds ex b4 being Linear_Combination of b3 st
      Carrier b4 <> {} & Sum b4 = 0. b2;

:: RANKNULL:funcnot 9 => RANKNULL:func 9
definition
  let a1 be non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
  let a2, a3 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over a1;
  let a4 be Element of bool the carrier of a2;
  let a5 be Function-like quasi_total linear Relation of the carrier of a2,the carrier of a3;
  let a6 be Linear_Combination of a5 .: a4;
  assume a5 | a4 is one-to-one;
  func A5 # A6 -> Linear_Combination of a4 equals
    (a6 * a5) +* ((a2 \ a4) --> 0. a1);
end;

:: RANKNULL:def 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, b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Element of bool the carrier of b2
for b5 being Function-like quasi_total linear Relation of the carrier of b2,the carrier of b3
for b6 being Linear_Combination of b5 .: b4
      st b5 | b4 is one-to-one
   holds b5 # b6 = (b6 * b5) +* ((b2 \ b4) --> 0. b1);

:: RANKNULL:th 42
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 non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total linear Relation of the carrier of b3,the carrier of b2
for b5 being Element of bool the carrier of b3
for b6 being Linear_Combination of b4 .: b5
for b7 being Element of the carrier of b3
      st b7 in b5 & b4 | b5 is one-to-one
   holds (b4 # b6) . b7 = b6 . (b4 . b7);

:: RANKNULL:th 43
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 non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total linear Relation of the carrier of b3,the carrier of b2
for b5 being Element of bool the carrier of b3
for b6 being Linear_Combination of b4 .: b5
      st b4 | b5 is one-to-one
   holds b4 @ (b4 # b6) = b6;

:: RANKNULL:funcnot 10 => RANKNULL:func 10
definition
  let a1 be non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
  let a2, a3 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like finite-dimensional VectSpStr over a1;
  let a4 be Function-like quasi_total linear Relation of the carrier of a2,the carrier of a3;
  func rank A4 -> natural set equals
    dim im a4;
end;

:: RANKNULL:def 7
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 non empty right_complementable Abelian add-associative right_zeroed VectSp-like finite-dimensional VectSpStr over b1
for b4 being Function-like quasi_total linear Relation of the carrier of b2,the carrier of b3 holds
   rank b4 = dim im b4;

:: RANKNULL:funcnot 11 => RANKNULL:func 11
definition
  let a1 be non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
  let a2, a3 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like finite-dimensional VectSpStr over a1;
  let a4 be Function-like quasi_total linear Relation of the carrier of a2,the carrier of a3;
  func nullity A4 -> natural set equals
    dim ker a4;
end;

:: RANKNULL:def 8
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 non empty right_complementable Abelian add-associative right_zeroed VectSp-like finite-dimensional VectSpStr over b1
for b4 being Function-like quasi_total linear Relation of the carrier of b2,the carrier of b3 holds
   nullity b4 = dim ker b4;

:: RANKNULL:th 44
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 non empty right_complementable Abelian add-associative right_zeroed VectSp-like finite-dimensional VectSpStr over b1
for b4 being Function-like quasi_total linear Relation of the carrier of b2,the carrier of b3 holds
   dim b2 = (rank b4) + nullity b4;

:: RANKNULL:th 45
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 non empty right_complementable Abelian add-associative right_zeroed VectSp-like finite-dimensional VectSpStr over b1
for b4 being Function-like quasi_total linear Relation of the carrier of b2,the carrier of b3
      st b4 is one-to-one
   holds dim b2 = rank b4;