Article MATRIX_8, MML version 4.99.1005

:: MATRIX_8:attrnot 1 => MATRIX_8:attr 1
definition
  let a1 be natural set;
  let a2 be non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr;
  let a3 be Matrix of a1,a1,the carrier of a2;
  attr a3 is Idempotent means
    a3 * a3 = a3;
end;

:: MATRIX_8:dfs 1
definiens
  let a1 be natural set;
  let a2 be non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr;
  let a3 be Matrix of a1,a1,the carrier of a2;
To prove
     a3 is Idempotent
it is sufficient to prove
  thus a3 * a3 = a3;

:: MATRIX_8:def 1
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3 being Matrix of b1,b1,the carrier of b2 holds
      b3 is Idempotent(b1, b2)
   iff
      b3 * b3 = b3;

:: MATRIX_8:attrnot 2 => MATRIX_8:attr 2
definition
  let a1 be natural set;
  let a2 be non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr;
  let a3 be Matrix of a1,a1,the carrier of a2;
  attr a3 is Nilpotent means
    a3 * a3 = 0.(a2,a1);
end;

:: MATRIX_8:dfs 2
definiens
  let a1 be natural set;
  let a2 be non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr;
  let a3 be Matrix of a1,a1,the carrier of a2;
To prove
     a3 is Nilpotent
it is sufficient to prove
  thus a3 * a3 = 0.(a2,a1);

:: MATRIX_8:def 2
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3 being Matrix of b1,b1,the carrier of b2 holds
      b3 is Nilpotent(b1, b2)
   iff
      b3 * b3 = 0.(b2,b1);

:: MATRIX_8:attrnot 3 => MATRIX_8:attr 3
definition
  let a1 be natural set;
  let a2 be non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr;
  let a3 be Matrix of a1,a1,the carrier of a2;
  attr a3 is Involutory means
    a3 * a3 = 1.(a2,a1);
end;

:: MATRIX_8:dfs 3
definiens
  let a1 be natural set;
  let a2 be non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr;
  let a3 be Matrix of a1,a1,the carrier of a2;
To prove
     a3 is Involutory
it is sufficient to prove
  thus a3 * a3 = 1.(a2,a1);

:: MATRIX_8:def 3
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3 being Matrix of b1,b1,the carrier of b2 holds
      b3 is Involutory(b1, b2)
   iff
      b3 * b3 = 1.(b2,b1);

:: MATRIX_8:attrnot 4 => MATRIX_8:attr 4
definition
  let a1 be natural set;
  let a2 be non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr;
  let a3 be Matrix of a1,a1,the carrier of a2;
  attr a3 is Self_Reversible means
    a3 is invertible(a1, a2) & a3 ~ = a3;
end;

:: MATRIX_8:dfs 4
definiens
  let a1 be natural set;
  let a2 be non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr;
  let a3 be Matrix of a1,a1,the carrier of a2;
To prove
     a3 is Self_Reversible
it is sufficient to prove
  thus a3 is invertible(a1, a2) & a3 ~ = a3;

:: MATRIX_8:def 4
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3 being Matrix of b1,b1,the carrier of b2 holds
      b3 is Self_Reversible(b1, b2)
   iff
      b3 is invertible(b1, b2) & b3 ~ = b3;

:: MATRIX_8:th 1
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr holds
   1.(b2,b1) is Idempotent(b1, b2) & 1.(b2,b1) is Involutory(b1, b2);

:: MATRIX_8:th 2
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr holds
   0.(b2,b1) is Idempotent(b1, b2) & 0.(b2,b1) is Nilpotent(b1, b2);

:: MATRIX_8:funcreg 1
registration
  let a1 be natural set;
  let a2 be non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr;
  cluster 1.(a2,a1) -> Idempotent Involutory;
end;

:: MATRIX_8:funcreg 2
registration
  let a1 be natural set;
  let a2 be non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr;
  cluster 0.(a2,a1) -> Idempotent Nilpotent;
end;

:: MATRIX_8:th 3
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3 being Matrix of b1,b1,the carrier of b2
      st 0 < b1
   holds    b3 is Idempotent(b1, b2)
   iff
      b3 @ is Idempotent(b1, b2);

:: MATRIX_8:th 4
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3 being Matrix of b1,b1,the carrier of b2
      st b3 is Involutory(b1, b2)
   holds b3 is invertible(b1, b2);

:: MATRIX_8:th 5
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3, b4 being Matrix of b1,b1,the carrier of b2
      st b3 is Idempotent(b1, b2) & b4 is Idempotent(b1, b2) & b3 commutes_with b4
   holds b3 * b3 commutes_with b4 * b4;

:: MATRIX_8:th 6
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3, b4 being Matrix of b1,b1,the carrier of b2
      st 0 < b1 & b3 is Idempotent(b1, b2) & b4 is Idempotent(b1, b2) & b3 commutes_with b4 & b3 * b4 = 0.(b2,b1)
   holds b3 + b4 is Idempotent(b1, b2);

:: MATRIX_8:th 7
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3, b4 being Matrix of b1,b1,the carrier of b2
      st b3 is Idempotent(b1, b2) & b4 is Idempotent(b1, b2) & b3 * b4 = - (b4 * b3)
   holds b3 + b4 is Idempotent(b1, b2);

:: MATRIX_8:th 8
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3, b4 being Matrix of b1,b1,the carrier of b2
      st b3 is Idempotent(b1, b2) & b4 is invertible(b1, b2)
   holds (b4 ~ * b3) * b4 is Idempotent(b1, b2);

:: MATRIX_8:th 9
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3 being Matrix of b1,b1,the carrier of b2
      st b3 is invertible(b1, b2) & b3 is Idempotent(b1, b2)
   holds b3 ~ is Idempotent(b1, b2);

:: MATRIX_8:th 10
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3 being Matrix of b1,b1,the carrier of b2
      st b3 is invertible(b1, b2) & b3 is Idempotent(b1, b2)
   holds b3 = 1.(b2,b1);

:: MATRIX_8:th 11
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3, b4 being Matrix of b1,b1,the carrier of b2
      st b3 is Idempotent(b1, b2) & b4 is Idempotent(b1, b2) & b3 commutes_with b4
   holds b3 * b4 is Idempotent(b1, b2);

:: MATRIX_8:th 12
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3, b4 being Matrix of b1,b1,the carrier of b2
      st b3 is Idempotent(b1, b2) & b4 is Idempotent(b1, b2) & b3 commutes_with b4
   holds b3 @ * (b4 @) is Idempotent(b1, b2);

:: MATRIX_8:th 13
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3, b4 being Matrix of b1,b1,the carrier of b2
      st b3 is Idempotent(b1, b2) & b4 is Idempotent(b1, b2) & b3 is invertible(b1, b2)
   holds b3 * b4 is Idempotent(b1, b2);

:: MATRIX_8:th 14
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3 being Matrix of b1,b1,the carrier of b2
      st b3 is Idempotent(b1, b2) & b3 is Orthogonal(b1, b2)
   holds b3 is symmetric(b1, b2);

:: MATRIX_8:th 15
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3, b4 being Matrix of b1,b1,the carrier of b2
      st b3 is Idempotent(b1, b2) & b4 is Idempotent(b1, b2) & b4 * b3 = 1.(b2,b1)
   holds b3 * b4 is Idempotent(b1, b2);

:: MATRIX_8:th 16
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3 being Matrix of b1,b1,the carrier of b2
      st b3 is Idempotent(b1, b2) & b3 is Orthogonal(b1, b2)
   holds b3 = 1.(b2,b1);

:: MATRIX_8:th 17
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3 being Matrix of b1,b1,the carrier of b2
      st b3 is symmetric(b1, b2)
   holds b3 * (b3 @) is symmetric(b1, b2);

:: MATRIX_8:th 18
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3 being Matrix of b1,b1,the carrier of b2
      st b3 is symmetric(b1, b2)
   holds b3 @ * b3 is symmetric(b1, b2);

:: MATRIX_8:th 19
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3, b4, b5 being Matrix of b1,b1,the carrier of b2
      st b3 is invertible(b1, b2) & b3 * b4 = b3 * b5
   holds b4 = b5;

:: MATRIX_8:th 20
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3, b4, b5 being Matrix of b1,b1,the carrier of b2
      st b3 is invertible(b1, b2) & b4 * b3 = b5 * b3
   holds b4 = b5;

:: MATRIX_8:th 21
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3, b4 being Matrix of b1,b1,the carrier of b2
      st b3 is invertible(b1, b2) & b4 * b3 = 0.(b2,b1)
   holds b4 = 0.(b2,b1);

:: MATRIX_8:th 22
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3, b4 being Matrix of b1,b1,the carrier of b2
      st b3 is invertible(b1, b2) & b4 * b3 = 0.(b2,b1)
   holds b4 = 0.(b2,b1);

:: MATRIX_8:th 23
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3, b4 being Matrix of b1,b1,the carrier of b2
      st b3 is Nilpotent(b1, b2) & b3 commutes_with b4 & 0 < b1
   holds b3 * b4 is Nilpotent(b1, b2);

:: MATRIX_8:th 24
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3, b4 being Matrix of b1,b1,the carrier of b2
      st 0 < b1 & b3 is Nilpotent(b1, b2) & b4 is Nilpotent(b1, b2) & b3 commutes_with b4 & b3 * b4 = 0.(b2,b1)
   holds b3 + b4 is Nilpotent(b1, b2);

:: MATRIX_8:th 25
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3, b4 being Matrix of b1,b1,the carrier of b2
      st b3 is Nilpotent(b1, b2) & b4 is Nilpotent(b1, b2) & b3 * b4 = - (b4 * b3) & 0 < b1
   holds b3 + b4 is Nilpotent(b1, b2);

:: MATRIX_8:th 26
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3 being Matrix of b1,b1,the carrier of b2
      st b3 is Nilpotent(b1, b2) & 0 < b1
   holds b3 @ is Nilpotent(b1, b2);

:: MATRIX_8:th 27
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3 being Matrix of b1,b1,the carrier of b2
      st b3 is Nilpotent(b1, b2) & b3 is Idempotent(b1, b2)
   holds b3 = 0.(b2,b1);

:: MATRIX_8:th 28
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
      st 0 < b1
   holds 0.(b2,b1) <> 1.(b2,b1);

:: MATRIX_8:th 29
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3 being Matrix of b1,b1,the carrier of b2
      st 0 < b1 & b3 is Nilpotent(b1, b2)
   holds b3 is not invertible(b1, b2);

:: MATRIX_8:th 30
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3 being Matrix of b1,b1,the carrier of b2
      st b3 is Self_Reversible(b1, b2)
   holds b3 is Involutory(b1, b2);

:: MATRIX_8:th 31
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr holds
   1.(b2,b1) is Self_Reversible(b1, b2);

:: MATRIX_8:th 32
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3 being Matrix of b1,b1,the carrier of b2
      st b3 is Self_Reversible(b1, b2) & b3 is Idempotent(b1, b2)
   holds b3 = 1.(b2,b1);

:: MATRIX_8:th 33
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3 being Matrix of b1,b1,the carrier of b2
      st b3 is Self_Reversible(b1, b2) & b3 is symmetric(b1, b2)
   holds b3 is Orthogonal(b1, b2);

:: MATRIX_8:prednot 1 => MATRIX_8:pred 1
definition
  let a1 be natural set;
  let a2 be non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr;
  let a3, a4 be Matrix of a1,a1,the carrier of a2;
  pred A3 is_similar_to A4 means
    ex b1 being Matrix of a1,a1,the carrier of a2 st
       b1 is invertible(a1, a2) & a3 = (b1 ~ * a4) * b1;
  symmetry;
::  for a1 being natural set
::  for a2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
::  for a3, a4 being Matrix of a1,a1,the carrier of a2
::        st a3 is_similar_to a4
::     holds a4 is_similar_to a3;
  reflexivity;
::  for a1 being natural set
::  for a2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
::  for a3 being Matrix of a1,a1,the carrier of a2 holds
::     a3 is_similar_to a3;
end;

:: MATRIX_8:dfs 5
definiens
  let a1 be natural set;
  let a2 be non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr;
  let a3, a4 be Matrix of a1,a1,the carrier of a2;
To prove
     a3 is_similar_to a4
it is sufficient to prove
  thus ex b1 being Matrix of a1,a1,the carrier of a2 st
       b1 is invertible(a1, a2) & a3 = (b1 ~ * a4) * b1;

:: MATRIX_8:def 5
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3, b4 being Matrix of b1,b1,the carrier of b2 holds
   b3 is_similar_to b4
iff
   ex b5 being Matrix of b1,b1,the carrier of b2 st
      b5 is invertible(b1, b2) & b3 = (b5 ~ * b4) * b5;

:: MATRIX_8:th 34
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3, b4, b5 being Matrix of b1,b1,the carrier of b2
      st b3 is_similar_to b4 & b4 is_similar_to b5
   holds b3 is_similar_to b5;

:: MATRIX_8:th 35
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3, b4 being Matrix of b1,b1,the carrier of b2
      st b3 is_similar_to b4 & b4 is Idempotent(b1, b2)
   holds b3 is Idempotent(b1, b2);

:: MATRIX_8:th 36
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3, b4 being Matrix of b1,b1,the carrier of b2
      st b3 is_similar_to b4 & b4 is Nilpotent(b1, b2) & 0 < b1
   holds b3 is Nilpotent(b1, b2);

:: MATRIX_8:th 37
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3, b4 being Matrix of b1,b1,the carrier of b2
      st b3 is_similar_to b4 & b4 is Involutory(b1, b2)
   holds b3 is Involutory(b1, b2);

:: MATRIX_8:th 38
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3, b4 being Matrix of b1,b1,the carrier of b2
      st b3 is_similar_to b4 & 0 < b1
   holds b3 + 1.(b2,b1) is_similar_to b4 + 1.(b2,b1);

:: MATRIX_8:th 39
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3, b4 being Matrix of b1,b1,the carrier of b2
      st b3 is_similar_to b4 & 0 < b1
   holds b3 + b3 is_similar_to b4 + b4;

:: MATRIX_8:th 40
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3, b4 being Matrix of b1,b1,the carrier of b2
      st b3 is_similar_to b4 & 0 < b1
   holds (b3 + b3) + b3 is_similar_to (b4 + b4) + b4;

:: MATRIX_8:th 41
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3, b4 being Matrix of b1,b1,the carrier of b2
      st b3 is invertible(b1, b2)
   holds b4 * b3 is_similar_to b3 * b4;

:: MATRIX_8:th 42
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3, b4 being Matrix of b1,b1,the carrier of b2
      st b3 is invertible(b1, b2) & b4 is_similar_to b3
   holds b4 is invertible(b1, b2);

:: MATRIX_8:th 43
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3, b4 being Matrix of b1,b1,the carrier of b2
      st b3 is invertible(b1, b2) & b4 is_similar_to b3
   holds b4 ~ is_similar_to b3 ~;

:: MATRIX_8:prednot 2 => MATRIX_8:pred 2
definition
  let a1 be natural set;
  let a2 be non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr;
  let a3, a4 be Matrix of a1,a1,the carrier of a2;
  pred A3 is_congruent_Matrix_of A4 means
    ex b1 being Matrix of a1,a1,the carrier of a2 st
       b1 is invertible(a1, a2) & a3 = (b1 @ * a4) * b1;
  reflexivity;
::  for a1 being natural set
::  for a2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
::  for a3 being Matrix of a1,a1,the carrier of a2 holds
::     a3 is_congruent_Matrix_of a3;
end;

:: MATRIX_8:dfs 6
definiens
  let a1 be natural set;
  let a2 be non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr;
  let a3, a4 be Matrix of a1,a1,the carrier of a2;
To prove
     a3 is_congruent_Matrix_of a4
it is sufficient to prove
  thus ex b1 being Matrix of a1,a1,the carrier of a2 st
       b1 is invertible(a1, a2) & a3 = (b1 @ * a4) * b1;

:: MATRIX_8:def 6
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3, b4 being Matrix of b1,b1,the carrier of b2 holds
   b3 is_congruent_Matrix_of b4
iff
   ex b5 being Matrix of b1,b1,the carrier of b2 st
      b5 is invertible(b1, b2) & b3 = (b5 @ * b4) * b5;

:: MATRIX_8:th 45
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3, b4 being Matrix of b1,b1,the carrier of b2
      st b3 is_congruent_Matrix_of b4 & 0 < b1
   holds b4 is_congruent_Matrix_of b3;

:: MATRIX_8:th 46
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3, b4, b5 being Matrix of b1,b1,the carrier of b2
      st b3 is_congruent_Matrix_of b4 & b4 is_congruent_Matrix_of b5 & 0 < b1
   holds b3 is_congruent_Matrix_of b5;

:: MATRIX_8:th 47
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3, b4 being Matrix of b1,b1,the carrier of b2
      st b3 is_congruent_Matrix_of b4 & 0 < b1
   holds b3 + b3 is_congruent_Matrix_of b4 + b4;

:: MATRIX_8:th 48
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3, b4 being Matrix of b1,b1,the carrier of b2
      st b3 is_congruent_Matrix_of b4 & 0 < b1
   holds (b3 + b3) + b3 is_congruent_Matrix_of (b4 + b4) + b4;

:: MATRIX_8:th 49
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3, b4 being Matrix of b1,b1,the carrier of b2
      st b3 is Orthogonal(b1, b2)
   holds b4 * b3 is_congruent_Matrix_of b3 * b4;

:: MATRIX_8:th 50
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3, b4 being Matrix of b1,b1,the carrier of b2
      st b3 is invertible(b1, b2) & b4 is_congruent_Matrix_of b3 & 0 < b1
   holds b4 is invertible(b1, b2);

:: MATRIX_8:th 51
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3, b4 being Matrix of b1,b1,the carrier of b2
      st b3 is invertible(b1, b2) & b4 is_congruent_Matrix_of b3 & 0 < b1
   holds b4 @ is_congruent_Matrix_of b3 @;

:: MATRIX_8:th 52
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3, b4 being Matrix of b1,b1,the carrier of b2
      st b3 is Orthogonal(b1, b2)
   holds (b3 @ * b4) * b3 is_similar_to b4;

:: MATRIX_8:funcnot 1 => MATRIX_8:func 1
definition
  let a1 be natural set;
  let a2 be non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr;
  let a3 be Matrix of a1,a1,the carrier of a2;
  func Trace A3 -> Element of the carrier of a2 equals
    Sum diagonal_of_Matrix a3;
end;

:: MATRIX_8:def 7
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3 being Matrix of b1,b1,the carrier of b2 holds
   Trace b3 = Sum diagonal_of_Matrix b3;

:: MATRIX_8:th 53
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3 being Matrix of b1,b1,the carrier of b2 holds
   Trace b3 = Trace (b3 @);

:: MATRIX_8:th 54
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3, b4 being Matrix of b1,b1,the carrier of b2 holds
Trace (b3 + b4) = (Trace b3) + Trace b4;

:: MATRIX_8:th 55
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3, b4, b5 being Matrix of b1,b1,the carrier of b2 holds
Trace ((b3 + b4) + b5) = ((Trace b3) + Trace b4) + Trace b5;

:: MATRIX_8:th 56
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr holds
   Trace 0.(b2,b1) = 0. b2;

:: MATRIX_8:th 57
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3 being Matrix of b1,b1,the carrier of b2 holds
   Trace - b3 = - Trace b3;

:: MATRIX_8:th 58
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3 being Matrix of b1,b1,the carrier of b2 holds
   - Trace - b3 = Trace b3;

:: MATRIX_8:th 59
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3 being Matrix of b1,b1,the carrier of b2 holds
   Trace (b3 + - b3) = 0. b2;

:: MATRIX_8:th 60
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3, b4 being Matrix of b1,b1,the carrier of b2 holds
Trace (b3 - b4) = (Trace b3) - Trace b4;

:: MATRIX_8:th 61
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3, b4, b5 being Matrix of b1,b1,the carrier of b2 holds
Trace ((b3 - b4) + b5) = ((Trace b3) - Trace b4) + Trace b5;

:: MATRIX_8:th 62
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3, b4, b5 being Matrix of b1,b1,the carrier of b2 holds
Trace ((b3 + b4) - b5) = ((Trace b3) + Trace b4) - Trace b5;

:: MATRIX_8:th 63
theorem
for b1 being natural set
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b3, b4, b5 being Matrix of b1,b1,the carrier of b2 holds
Trace ((b3 - b4) - b5) = ((Trace b3) - Trace b4) - Trace b5;