Article MATRIX_4, MML version 4.99.1005
:: MATRIX_4:funcnot 1 => MATRIX_4:func 1
definition
let a1 be non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr;
let a2, a3 be tabular FinSequence of (the carrier of a1) *;
func A2 - A3 -> tabular FinSequence of (the carrier of a1) * equals
a2 + - a3;
end;
:: MATRIX_4:def 1
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3 being tabular FinSequence of (the carrier of b1) * holds
b2 - b3 = b2 + - b3;
:: MATRIX_4:th 1
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2 being tabular FinSequence of (the carrier of b1) * holds
- - b2 = b2;
:: MATRIX_4:th 2
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2 being tabular FinSequence of (the carrier of b1) * holds
b2 + - b2 = 0.(b1,len b2,width b2);
:: MATRIX_4:th 3
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2 being tabular FinSequence of (the carrier of b1) * holds
b2 - b2 = 0.(b1,len b2,width b2);
:: MATRIX_4:th 4
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3, b4 being tabular FinSequence of (the carrier of b1) *
st len b2 = len b3 & len b3 = len b4 & width b2 = width b3 & width b3 = width b4 & b2 + b4 = b3 + b4
holds b2 = b3;
:: MATRIX_4:th 5
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3 being tabular FinSequence of (the carrier of b1) * holds
b2 - - b3 = b2 + b3;
:: MATRIX_4:th 6
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3 being tabular FinSequence of (the carrier of b1) *
st len b2 = len b3 & width b2 = width b3 & b2 = b2 + b3
holds b3 = 0.(b1,len b2,width b2);
:: MATRIX_4:th 7
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3 being tabular FinSequence of (the carrier of b1) *
st len b2 = len b3 &
width b2 = width b3 &
b2 - b3 = 0.(b1,len b2,width b2)
holds b2 = b3;
:: MATRIX_4:th 8
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3 being tabular FinSequence of (the carrier of b1) *
st len b2 = len b3 &
width b2 = width b3 &
b2 + b3 = 0.(b1,len b2,width b2)
holds b3 = - b2;
:: MATRIX_4:th 9
theorem
for b1, b2 being Element of NAT
for b3 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr holds
- 0.(b3,b1,b2) = 0.(b3,b1,b2);
:: MATRIX_4:th 10
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3 being tabular FinSequence of (the carrier of b1) *
st len b2 = len b3 & width b2 = width b3 & b3 - b2 = b3
holds b2 = 0.(b1,len b2,width b2);
:: MATRIX_4:th 11
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3 being tabular FinSequence of (the carrier of b1) *
st len b2 = len b3 & width b2 = width b3
holds b2 = b2 - (b3 - b3);
:: MATRIX_4:th 12
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3 being tabular FinSequence of (the carrier of b1) *
st len b2 = len b3 & width b2 = width b3
holds - (b2 + b3) = (- b2) + - b3;
:: MATRIX_4:th 13
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3 being tabular FinSequence of (the carrier of b1) *
st len b2 = len b3 & width b2 = width b3
holds b2 - (b2 - b3) = b3;
:: MATRIX_4:th 14
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3, b4 being tabular FinSequence of (the carrier of b1) *
st len b2 = len b3 & len b3 = len b4 & width b2 = width b3 & width b3 = width b4 & b2 - b4 = b3 - b4
holds b2 = b3;
:: MATRIX_4:th 15
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3, b4 being tabular FinSequence of (the carrier of b1) *
st len b2 = len b3 & len b3 = len b4 & width b2 = width b3 & width b3 = width b4 & b4 - b2 = b4 - b3
holds b2 = b3;
:: MATRIX_4:th 16
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3, b4 being tabular FinSequence of (the carrier of b1) *
st len b2 = len b3 & len b3 = len b4 & width b2 = width b3 & width b3 = width b4
holds (b2 - b3) - b4 = (b2 - b4) - b3;
:: MATRIX_4:th 17
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3, b4 being tabular FinSequence of (the carrier of b1) *
st len b2 = len b3 & len b3 = len b4 & width b2 = width b3 & width b3 = width b4
holds b2 - b4 = (b2 - b3) - (b4 - b3);
:: MATRIX_4:th 18
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3, b4 being tabular FinSequence of (the carrier of b1) *
st len b2 = len b3 & len b3 = len b4 & width b2 = width b3 & width b3 = width b4
holds (b4 - b2) - (b4 - b3) = b3 - b2;
:: MATRIX_4:th 19
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3, b4, b5 being tabular FinSequence of (the carrier of b1) *
st len b2 = len b3 & len b3 = len b4 & len b4 = len b5 & width b2 = width b3 & width b3 = width b4 & width b4 = width b5 & b2 - b3 = b4 - b5
holds b2 - b4 = b3 - b5;
:: MATRIX_4:th 20
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3 being tabular FinSequence of (the carrier of b1) *
st len b2 = len b3 & width b2 = width b3
holds b2 = b2 + (b3 - b3);
:: MATRIX_4:th 21
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3 being tabular FinSequence of (the carrier of b1) *
st len b2 = len b3 & width b2 = width b3
holds b2 = (b2 + b3) - b3;
:: MATRIX_4:th 22
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3 being tabular FinSequence of (the carrier of b1) *
st len b2 = len b3 & width b2 = width b3
holds b2 = (b2 - b3) + b3;
:: MATRIX_4:th 23
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3, b4 being tabular FinSequence of (the carrier of b1) *
st len b2 = len b3 & len b3 = len b4 & width b2 = width b3 & width b3 = width b4
holds b2 + b4 = (b2 + b3) + (b4 - b3);
:: MATRIX_4:th 24
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3, b4 being tabular FinSequence of (the carrier of b1) *
st len b2 = len b3 & len b3 = len b4 & width b2 = width b3 & width b3 = width b4
holds (b2 + b3) - b4 = (b2 - b4) + b3;
:: MATRIX_4:th 25
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3, b4 being tabular FinSequence of (the carrier of b1) *
st len b2 = len b3 & len b3 = len b4 & width b2 = width b3 & width b3 = width b4
holds (b2 - b3) + b4 = (b4 - b3) + b2;
:: MATRIX_4:th 26
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3, b4 being tabular FinSequence of (the carrier of b1) *
st len b2 = len b3 & len b3 = len b4 & width b2 = width b3 & width b3 = width b4
holds b2 + b4 = (b2 + b3) - (b3 - b4);
:: MATRIX_4:th 27
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3, b4 being tabular FinSequence of (the carrier of b1) *
st len b2 = len b3 & len b3 = len b4 & width b2 = width b3 & width b3 = width b4
holds b2 - b4 = (b2 + b3) - (b4 + b3);
:: MATRIX_4:th 28
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3, b4, b5 being tabular FinSequence of (the carrier of b1) *
st len b2 = len b3 & len b3 = len b4 & len b4 = len b5 & width b2 = width b3 & width b3 = width b4 & width b4 = width b5 & b2 + b3 = b4 + b5
holds b2 - b4 = b5 - b3;
:: MATRIX_4:th 29
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3, b4, b5 being tabular FinSequence of (the carrier of b1) *
st len b2 = len b3 & len b3 = len b4 & len b4 = len b5 & width b2 = width b3 & width b3 = width b4 & width b4 = width b5 & b2 - b4 = b5 - b3
holds b2 + b3 = b4 + b5;
:: MATRIX_4:th 30
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3, b4, b5 being tabular FinSequence of (the carrier of b1) *
st len b2 = len b3 & len b3 = len b4 & len b4 = len b5 & width b2 = width b3 & width b3 = width b4 & width b4 = width b5 & b2 + b3 = b4 - b5
holds b2 + b5 = b4 - b3;
:: MATRIX_4:th 31
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3, b4 being tabular FinSequence of (the carrier of b1) *
st len b2 = len b3 & len b3 = len b4 & width b2 = width b3 & width b3 = width b4
holds b2 - (b3 + b4) = (b2 - b3) - b4;
:: MATRIX_4:th 32
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3, b4 being tabular FinSequence of (the carrier of b1) *
st len b2 = len b3 & len b3 = len b4 & width b2 = width b3 & width b3 = width b4
holds b2 - (b3 - b4) = (b2 - b3) + b4;
:: MATRIX_4:th 33
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3, b4 being tabular FinSequence of (the carrier of b1) *
st len b2 = len b3 & len b3 = len b4 & width b2 = width b3 & width b3 = width b4
holds b2 - (b3 - b4) = b2 + (b4 - b3);
:: MATRIX_4:th 34
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3, b4 being tabular FinSequence of (the carrier of b1) *
st len b2 = len b3 & len b3 = len b4 & width b2 = width b3 & width b3 = width b4
holds b2 - b4 = (b2 - b3) + (b3 - b4);
:: MATRIX_4:th 35
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3, b4 being tabular FinSequence of (the carrier of b1) *
st len b2 = len b3 & len b3 = len b4 & width b2 = width b3 & width b3 = width b4 & - b2 = - b3
holds b2 = b3;
:: MATRIX_4:th 36
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2 being tabular FinSequence of (the carrier of b1) *
st - b2 = 0.(b1,len b2,width b2)
holds b2 = 0.(b1,len b2,width b2);
:: MATRIX_4:th 37
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3 being tabular FinSequence of (the carrier of b1) *
st len b2 = len b3 &
width b2 = width b3 &
b2 + - b3 = 0.(b1,len b2,width b2)
holds b2 = b3;
:: MATRIX_4:th 38
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3 being tabular FinSequence of (the carrier of b1) *
st len b2 = len b3 & width b2 = width b3
holds b2 = (b2 + b3) + - b3;
:: MATRIX_4:th 39
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3 being tabular FinSequence of (the carrier of b1) *
st len b2 = len b3 & width b2 = width b3
holds b2 = b2 + (b3 + - b3);
:: MATRIX_4:th 40
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3 being tabular FinSequence of (the carrier of b1) *
st len b2 = len b3 & width b2 = width b3
holds b2 = ((- b3) + b2) + b3;
:: MATRIX_4:th 41
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3 being tabular FinSequence of (the carrier of b1) *
st len b2 = len b3 & width b2 = width b3
holds - ((- b2) + b3) = b2 + - b3;
:: MATRIX_4:th 42
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3 being tabular FinSequence of (the carrier of b1) *
st len b2 = len b3 & width b2 = width b3
holds b2 + b3 = - ((- b2) + - b3);
:: MATRIX_4:th 43
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3 being tabular FinSequence of (the carrier of b1) *
st len b2 = len b3 & width b2 = width b3
holds - (b2 - b3) = b3 - b2;
:: MATRIX_4:th 44
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3 being tabular FinSequence of (the carrier of b1) *
st len b2 = len b3 & width b2 = width b3
holds (- b2) - b3 = (- b3) - b2;
:: MATRIX_4:th 45
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3 being tabular FinSequence of (the carrier of b1) *
st len b2 = len b3 & width b2 = width b3
holds b2 = (- b3) - ((- b2) - b3);
:: MATRIX_4:th 46
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3, b4 being tabular FinSequence of (the carrier of b1) *
st len b2 = len b3 & len b3 = len b4 & width b2 = width b3 & width b3 = width b4
holds ((- b2) - b3) - b4 = ((- b2) - b4) - b3;
:: MATRIX_4:th 47
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3, b4 being tabular FinSequence of (the carrier of b1) *
st len b2 = len b3 & len b3 = len b4 & width b2 = width b3 & width b3 = width b4
holds ((- b2) - b3) - b4 = ((- b3) - b4) - b2;
:: MATRIX_4:th 48
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3, b4 being tabular FinSequence of (the carrier of b1) *
st len b2 = len b3 & len b3 = len b4 & width b2 = width b3 & width b3 = width b4
holds ((- b2) - b3) - b4 = ((- b4) - b3) - b2;
:: MATRIX_4:th 49
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3, b4 being tabular FinSequence of (the carrier of b1) *
st len b2 = len b3 & len b3 = len b4 & width b2 = width b3 & width b3 = width b4
holds (b4 - b2) - (b4 - b3) = - (b2 - b3);
:: MATRIX_4:th 50
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2 being tabular FinSequence of (the carrier of b1) * holds
(0.(b1,len b2,width b2)) - b2 = - b2;
:: MATRIX_4:th 51
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3 being tabular FinSequence of (the carrier of b1) *
st len b2 = len b3 & width b2 = width b3
holds b2 + b3 = b2 - - b3;
:: MATRIX_4:th 52
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3 being tabular FinSequence of (the carrier of b1) *
st len b2 = len b3 & width b2 = width b3
holds b2 = b2 - (b3 + - b3);
:: MATRIX_4:th 53
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3, b4 being tabular FinSequence of (the carrier of b1) *
st len b2 = len b3 & len b3 = len b4 & width b2 = width b3 & width b3 = width b4 & b2 - b4 = b3 + - b4
holds b2 = b3;
:: MATRIX_4:th 54
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3, b4 being tabular FinSequence of (the carrier of b1) *
st len b2 = len b3 & len b3 = len b4 & width b2 = width b3 & width b3 = width b4 & b4 - b2 = b4 + - b3
holds b2 = b3;
:: MATRIX_4:th 55
theorem
for b1 being set
for b2, b3 being tabular FinSequence of b1 *
st len b2 = len b3 & width b2 = width b3
holds Indices b2 = Indices b3;
:: MATRIX_4:th 56
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3, b4 being FinSequence of the carrier of b1
st len b2 = len b3 & len b3 = len b4
holds mlt(b2 + b3,b4) = (mlt(b2,b4)) + mlt(b3,b4);
:: MATRIX_4:th 57
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3, b4 being FinSequence of the carrier of b1
st len b2 = len b3 & len b3 = len b4
holds mlt(b4,b2 + b3) = (mlt(b4,b2)) + mlt(b4,b3);
:: MATRIX_4:th 58
theorem
for b1 being non empty set
for b2 being tabular FinSequence of b1 *
st 0 < len b2
for b3 being Element of NAT holds
b2 is Matrix of b3,width b2,b1
iff
b3 = len b2;
:: MATRIX_4: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 Element of NAT
for b4, b5 being tabular FinSequence of (the carrier of b2) *
st len b4 = len b5 &
width b4 = width b5 &
(ex b6 being Element of NAT st
[b1,b6] in Indices b4)
holds Line(b4 + b5,b1) = (Line(b4,b1)) + Line(b5,b1);
:: MATRIX_4:th 60
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2 being natural set
for b3, b4 being tabular FinSequence of (the carrier of b1) *
st len b3 = len b4 &
width b3 = width b4 &
(ex b5 being natural set st
[b5,b2] in Indices b3)
holds Col(b3 + b4,b2) = (Col(b3,b2)) + Col(b4,b2);
:: MATRIX_4:th 61
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3 being FinSequence of the carrier of b1
st len b2 = len b3
holds Sum (b2 + b3) = (Sum b2) + Sum b3;
:: MATRIX_4:th 62
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3, b4 being tabular FinSequence of (the carrier of b1) *
st len b3 = len b4 & width b3 = width b4 & width b2 = len b3 & 0 < len b2 & 0 < len b3
holds b2 * (b3 + b4) = (b2 * b3) + (b2 * b4);
:: MATRIX_4:th 63
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3, b4 being tabular FinSequence of (the carrier of b1) *
st len b3 = len b4 & width b3 = width b4 & len b2 = width b3 & 0 < len b3 & 0 < len b2
holds (b3 + b4) * b2 = (b3 * b2) + (b4 * b2);
:: MATRIX_4:th 64
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible associative commutative Abelian add-associative right_zeroed well-unital distributive doubleLoopStr
for b2, b3, b4 being Element of NAT
for b5 being Matrix of b2,b3,the carrier of b1
for b6 being Matrix of b3,b4,the carrier of b1
st width b5 = len b6 & 0 < len b5 & 0 < len b6
holds b5 * b6 is Matrix of b2,b4,the carrier of b1;