Article MATRIX10, MML version 4.99.1005
:: MATRIX10:attrnot 1 => MATRIX10:attr 1
definition
let a1 be tabular FinSequence of REAL *;
attr a1 is Positive means
for b1, b2 being natural set
st [b1,b2] in Indices a1
holds 0 < a1 *(b1,b2);
end;
:: MATRIX10:dfs 1
definiens
let a1 be tabular FinSequence of REAL *;
To prove
a1 is Positive
it is sufficient to prove
thus for b1, b2 being natural set
st [b1,b2] in Indices a1
holds 0 < a1 *(b1,b2);
:: MATRIX10:def 1
theorem
for b1 being tabular FinSequence of REAL * holds
b1 is Positive
iff
for b2, b3 being natural set
st [b2,b3] in Indices b1
holds 0 < b1 *(b2,b3);
:: MATRIX10:attrnot 2 => MATRIX10:attr 2
definition
let a1 be tabular FinSequence of REAL *;
attr a1 is Negative means
for b1, b2 being natural set
st [b1,b2] in Indices a1
holds a1 *(b1,b2) < 0;
end;
:: MATRIX10:dfs 2
definiens
let a1 be tabular FinSequence of REAL *;
To prove
a1 is Negative
it is sufficient to prove
thus for b1, b2 being natural set
st [b1,b2] in Indices a1
holds a1 *(b1,b2) < 0;
:: MATRIX10:def 2
theorem
for b1 being tabular FinSequence of REAL * holds
b1 is Negative
iff
for b2, b3 being natural set
st [b2,b3] in Indices b1
holds b1 *(b2,b3) < 0;
:: MATRIX10:attrnot 3 => MATRIX10:attr 3
definition
let a1 be tabular FinSequence of REAL *;
attr a1 is Nonpositive means
for b1, b2 being natural set
st [b1,b2] in Indices a1
holds a1 *(b1,b2) <= 0;
end;
:: MATRIX10:dfs 3
definiens
let a1 be tabular FinSequence of REAL *;
To prove
a1 is Nonpositive
it is sufficient to prove
thus for b1, b2 being natural set
st [b1,b2] in Indices a1
holds a1 *(b1,b2) <= 0;
:: MATRIX10:def 3
theorem
for b1 being tabular FinSequence of REAL * holds
b1 is Nonpositive
iff
for b2, b3 being natural set
st [b2,b3] in Indices b1
holds b1 *(b2,b3) <= 0;
:: MATRIX10:attrnot 4 => MATRIX10:attr 4
definition
let a1 be tabular FinSequence of REAL *;
attr a1 is Nonnegative means
for b1, b2 being natural set
st [b1,b2] in Indices a1
holds 0 <= a1 *(b1,b2);
end;
:: MATRIX10:dfs 4
definiens
let a1 be tabular FinSequence of REAL *;
To prove
a1 is Nonnegative
it is sufficient to prove
thus for b1, b2 being natural set
st [b1,b2] in Indices a1
holds 0 <= a1 *(b1,b2);
:: MATRIX10:def 4
theorem
for b1 being tabular FinSequence of REAL * holds
b1 is Nonnegative
iff
for b2, b3 being natural set
st [b2,b3] in Indices b1
holds 0 <= b1 *(b2,b3);
:: MATRIX10:prednot 1 => MATRIX10:pred 1
definition
let a1, a2 be tabular FinSequence of REAL *;
pred A1 is_less_than A2 means
for b1, b2 being natural set
st [b1,b2] in Indices a1
holds a1 *(b1,b2) < a2 *(b1,b2);
end;
:: MATRIX10:dfs 5
definiens
let a1, a2 be tabular FinSequence of REAL *;
To prove
a1 is_less_than a2
it is sufficient to prove
thus for b1, b2 being natural set
st [b1,b2] in Indices a1
holds a1 *(b1,b2) < a2 *(b1,b2);
:: MATRIX10:def 5
theorem
for b1, b2 being tabular FinSequence of REAL * holds
b1 is_less_than b2
iff
for b3, b4 being natural set
st [b3,b4] in Indices b1
holds b1 *(b3,b4) < b2 *(b3,b4);
:: MATRIX10:prednot 2 => MATRIX10:pred 2
definition
let a1, a2 be tabular FinSequence of REAL *;
pred A1 is_less_or_equal_with A2 means
for b1, b2 being natural set
st [b1,b2] in Indices a1
holds a1 *(b1,b2) <= a2 *(b1,b2);
end;
:: MATRIX10:dfs 6
definiens
let a1, a2 be tabular FinSequence of REAL *;
To prove
a1 is_less_or_equal_with a2
it is sufficient to prove
thus for b1, b2 being natural set
st [b1,b2] in Indices a1
holds a1 *(b1,b2) <= a2 *(b1,b2);
:: MATRIX10:def 6
theorem
for b1, b2 being tabular FinSequence of REAL * holds
b1 is_less_or_equal_with b2
iff
for b3, b4 being natural set
st [b3,b4] in Indices b1
holds b1 *(b3,b4) <= b2 *(b3,b4);
:: MATRIX10:funcnot 1 => MATRIX10:func 1
definition
let a1 be tabular FinSequence of REAL *;
func |:A1:| -> tabular FinSequence of REAL * means
len it = len a1 &
width it = width a1 &
(for b1, b2 being natural set
st [b1,b2] in Indices a1
holds it *(b1,b2) = abs (a1 *(b1,b2)));
end;
:: MATRIX10:def 7
theorem
for b1, b2 being tabular FinSequence of REAL * holds
b2 = |:b1:|
iff
len b2 = len b1 &
width b2 = width b1 &
(for b3, b4 being natural set
st [b3,b4] in Indices b1
holds b2 *(b3,b4) = abs (b1 *(b3,b4)));
:: MATRIX10:funcnot 2 => MATRIX10:func 2
definition
let a1 be natural set;
let a2 be Matrix of a1,a1,REAL;
redefine func - a2 -> Matrix of a1,a1,REAL;
end;
:: MATRIX10:funcnot 3 => MATRIX10:func 3
definition
let a1 be natural set;
let a2, a3 be Matrix of a1,a1,REAL;
redefine func a2 + a3 -> Matrix of a1,a1,REAL;
end;
:: MATRIX10:funcnot 4 => MATRIX10:func 4
definition
let a1 be natural set;
let a2, a3 be Matrix of a1,a1,REAL;
redefine func a2 - a3 -> Matrix of a1,a1,REAL;
end;
:: MATRIX10:funcnot 5 => MATRIX10:func 5
definition
let a1 be natural set;
let a2 be Element of REAL;
let a3 be Matrix of a1,a1,REAL;
redefine func a2 * a3 -> Matrix of a1,a1,REAL;
end;
:: MATRIX10:funcreg 1
registration
cluster (1,1)--> 1 -> tabular Positive;
end;
:: MATRIX10:funcreg 2
registration
cluster (1,1)--> 1 -> tabular Nonnegative;
end;
:: MATRIX10:funcreg 3
registration
cluster (1,1)--> - 1 -> tabular Negative;
end;
:: MATRIX10:funcreg 4
registration
cluster (1,1)--> - 1 -> tabular Nonpositive;
end;
:: MATRIX10:exreg 1
registration
cluster Relation-like Function-like FinSequence-like tabular Positive Nonnegative FinSequence of REAL *;
end;
:: MATRIX10:exreg 2
registration
cluster Relation-like Function-like FinSequence-like tabular Negative Nonpositive FinSequence of REAL *;
end;
:: MATRIX10:funcreg 5
registration
let a1 be tabular Positive FinSequence of REAL *;
cluster a1 @ -> tabular Positive;
end;
:: MATRIX10:funcreg 6
registration
let a1 be tabular Negative FinSequence of REAL *;
cluster a1 @ -> tabular Negative;
end;
:: MATRIX10:funcreg 7
registration
let a1 be tabular Nonpositive FinSequence of REAL *;
cluster a1 @ -> tabular Nonpositive;
end;
:: MATRIX10:funcreg 8
registration
let a1 be tabular Nonnegative FinSequence of REAL *;
cluster a1 @ -> tabular Nonnegative;
end;
:: MATRIX10:funcreg 9
registration
let a1 be natural set;
cluster (a1,a1)--> 1 -> Positive;
end;
:: MATRIX10:funcreg 10
registration
let a1 be natural set;
cluster (a1,a1)--> - 1 -> Negative;
end;
:: MATRIX10:funcreg 11
registration
let a1 be natural set;
cluster (a1,a1)--> - 1 -> Nonpositive;
end;
:: MATRIX10:funcreg 12
registration
let a1 be natural set;
cluster (a1,a1)--> 1 -> Nonnegative;
end;
:: MATRIX10:exreg 3
registration
let a1 be natural set;
cluster Relation-like Function-like FinSequence-like tabular Positive Nonnegative Matrix of a1,a1,REAL;
end;
:: MATRIX10:exreg 4
registration
let a1 be natural set;
cluster Relation-like Function-like FinSequence-like tabular Negative Nonpositive Matrix of a1,a1,REAL;
end;
:: MATRIX10:th 1
theorem
for b1 being Element of the carrier of F_Real
for b2 being Element of REAL
st b1 = b2
holds - b1 = - b2;
:: MATRIX10:th 2
theorem
for b1, b2 being natural set
for b3 being tabular FinSequence of REAL *
st [b1,b2] in Indices b3
holds (- b3) *(b1,b2) = - (b3 *(b1,b2));
:: MATRIX10:th 3
theorem
for b1, b2 being natural set
for b3, b4 being tabular FinSequence of REAL *
st len b3 = len b4 & width b3 = width b4 & [b1,b2] in Indices b3
holds (b3 - b4) *(b1,b2) = (b3 *(b1,b2)) - (b4 *(b1,b2));
:: MATRIX10:th 4
theorem
for b1 being Element of REAL
for b2, b3 being natural set
for b4 being tabular FinSequence of REAL *
st len (b1 * b4) = len b4 & width (b1 * b4) = width b4 & [b2,b3] in Indices b4
holds (b1 * b4) *(b2,b3) = b1 * (b4 *(b2,b3));
:: MATRIX10:th 5
theorem
for b1 being natural set
for b2 being Matrix of b1,b1,REAL holds
Indices b2 = Indices |:b2:|;
:: MATRIX10:th 6
theorem
for b1 being Element of REAL
for b2 being natural set
for b3 being Matrix of b2,b2,REAL holds
|:b1 * b3:| = (abs b1) * |:b3:|;
:: MATRIX10:th 7
theorem
for b1 being natural set
for b2 being Matrix of b1,b1,REAL
st b2 is Negative
holds - b2 is Positive;
:: MATRIX10:th 8
theorem
for b1 being natural set
for b2, b3 being Matrix of b1,b1,REAL
st b2 is Positive & b3 is Positive
holds b2 + b3 is Positive;
:: MATRIX10:th 9
theorem
for b1 being natural set
for b2, b3 being Matrix of b1,b1,REAL
st - b2 is_less_than b3
holds b3 + b2 is Positive;
:: MATRIX10:th 10
theorem
for b1 being natural set
for b2, b3 being Matrix of b1,b1,REAL
st b2 is Nonnegative & b3 is Positive
holds b2 + b3 is Positive;
:: MATRIX10:th 11
theorem
for b1 being natural set
for b2, b3 being Matrix of b1,b1,REAL
st b2 is Positive & b3 is Negative & |:b3:| is_less_than |:b2:|
holds b2 + b3 is Positive;
:: MATRIX10:th 12
theorem
for b1 being natural set
for b2, b3 being Matrix of b1,b1,REAL
st b2 is Positive & b3 is Negative
holds b2 - b3 is Positive;
:: MATRIX10:th 13
theorem
for b1 being natural set
for b2, b3 being Matrix of b1,b1,REAL
st b2 is_less_than b3
holds b3 - b2 is Positive;
:: MATRIX10:th 14
theorem
for b1 being Element of REAL
for b2 being natural set
for b3 being Matrix of b2,b2,REAL
st 0 < b1 & b3 is Positive
holds b1 * b3 is Positive;
:: MATRIX10:th 15
theorem
for b1 being Element of REAL
for b2 being natural set
for b3 being Matrix of b2,b2,REAL
st b1 < 0 & b3 is Negative
holds b1 * b3 is Positive;
:: MATRIX10:th 16
theorem
for b1 being natural set
for b2 being Matrix of b1,b1,REAL
st b2 is Positive
holds - b2 is Negative;
:: MATRIX10:th 17
theorem
for b1 being natural set
for b2, b3 being Matrix of b1,b1,REAL
st b2 is Negative & b3 is Negative
holds b2 + b3 is Negative;
:: MATRIX10:th 18
theorem
for b1 being natural set
for b2, b3 being Matrix of b1,b1,REAL
st b2 is_less_than - b3
holds b2 + b3 is Negative;
:: MATRIX10:th 19
theorem
for b1 being natural set
for b2, b3 being Matrix of b1,b1,REAL
st b2 is Positive & b3 is Negative & |:b2:| is_less_than |:b3:|
holds b2 + b3 is Negative;
:: MATRIX10:th 20
theorem
for b1 being natural set
for b2, b3 being Matrix of b1,b1,REAL
st b2 is_less_than b3
holds b2 - b3 is Negative;
:: MATRIX10:th 21
theorem
for b1 being natural set
for b2, b3 being Matrix of b1,b1,REAL
st b2 is Positive & b3 is Negative
holds b3 - b2 is Negative;
:: MATRIX10:th 22
theorem
for b1 being Element of REAL
for b2 being natural set
for b3 being Matrix of b2,b2,REAL
st b1 < 0 & b3 is Positive
holds b1 * b3 is Negative;
:: MATRIX10:th 23
theorem
for b1 being Element of REAL
for b2 being natural set
for b3 being Matrix of b2,b2,REAL
st 0 < b1 & b3 is Negative
holds b1 * b3 is Negative;
:: MATRIX10:th 24
theorem
for b1 being natural set
for b2 being Matrix of b1,b1,REAL
st b2 is Nonnegative
holds - b2 is Nonpositive;
:: MATRIX10:th 25
theorem
for b1 being natural set
for b2 being Matrix of b1,b1,REAL
st b2 is Negative
holds b2 is Nonpositive;
:: MATRIX10:th 26
theorem
for b1 being natural set
for b2, b3 being Matrix of b1,b1,REAL
st b2 is Nonpositive & b3 is Nonpositive
holds b2 + b3 is Nonpositive;
:: MATRIX10:th 27
theorem
for b1 being natural set
for b2, b3 being Matrix of b1,b1,REAL
st b2 is_less_or_equal_with - b3
holds b2 + b3 is Nonpositive;
:: MATRIX10:th 28
theorem
for b1 being natural set
for b2, b3 being Matrix of b1,b1,REAL
st b2 is_less_or_equal_with b3
holds b2 - b3 is Nonpositive;
:: MATRIX10:th 29
theorem
for b1 being Element of REAL
for b2 being natural set
for b3 being Matrix of b2,b2,REAL
st b1 <= 0 & b3 is Positive
holds b1 * b3 is Nonpositive;
:: MATRIX10:th 30
theorem
for b1 being Element of REAL
for b2 being natural set
for b3 being Matrix of b2,b2,REAL
st 0 <= b1 & b3 is Negative
holds b1 * b3 is Nonpositive;
:: MATRIX10:th 31
theorem
for b1 being Element of REAL
for b2 being natural set
for b3 being Matrix of b2,b2,REAL
st 0 <= b1 & b3 is Nonpositive
holds b1 * b3 is Nonpositive;
:: MATRIX10:th 32
theorem
for b1 being Element of REAL
for b2 being natural set
for b3 being Matrix of b2,b2,REAL
st b1 <= 0 & b3 is Nonnegative
holds b1 * b3 is Nonpositive;
:: MATRIX10:th 33
theorem
for b1 being natural set
for b2 being Matrix of b1,b1,REAL holds
|:b2:| is Nonnegative;
:: MATRIX10:th 34
theorem
for b1 being natural set
for b2 being Matrix of b1,b1,REAL
st b2 is Positive
holds b2 is Nonnegative;
:: MATRIX10:th 35
theorem
for b1 being natural set
for b2 being Matrix of b1,b1,REAL
st b2 is Nonpositive
holds - b2 is Nonnegative;
:: MATRIX10:th 36
theorem
for b1 being natural set
for b2, b3 being Matrix of b1,b1,REAL
st b2 is Nonnegative & b3 is Nonnegative
holds b2 + b3 is Nonnegative;
:: MATRIX10:th 37
theorem
for b1 being natural set
for b2, b3 being Matrix of b1,b1,REAL
st - b2 is_less_or_equal_with b3
holds b2 + b3 is Nonnegative;
:: MATRIX10:th 38
theorem
for b1 being natural set
for b2, b3 being Matrix of b1,b1,REAL
st b2 is_less_or_equal_with b3
holds b3 - b2 is Nonnegative;
:: MATRIX10:th 39
theorem
for b1 being Element of REAL
for b2 being natural set
for b3 being Matrix of b2,b2,REAL
st 0 <= b1 & b3 is Positive
holds b1 * b3 is Nonnegative;
:: MATRIX10:th 40
theorem
for b1 being Element of REAL
for b2 being natural set
for b3 being Matrix of b2,b2,REAL
st b1 <= 0 & b3 is Negative
holds b1 * b3 is Nonnegative;
:: MATRIX10:th 41
theorem
for b1 being Element of REAL
for b2 being natural set
for b3 being Matrix of b2,b2,REAL
st b1 <= 0 & b3 is Nonpositive
holds b1 * b3 is Nonnegative;
:: MATRIX10:th 42
theorem
for b1 being Element of REAL
for b2 being natural set
for b3 being Matrix of b2,b2,REAL
st 0 <= b1 & b3 is Nonnegative
holds b1 * b3 is Nonnegative;
:: MATRIX10:th 43
theorem
for b1, b2 being Element of REAL
for b3 being natural set
for b4, b5 being Matrix of b3,b3,REAL
st 0 <= b1 & 0 <= b2 & b4 is Nonnegative & b5 is Nonnegative
holds (b1 * b4) + (b2 * b5) is Nonnegative;
:: MATRIX10:th 44
theorem
for b1 being natural set
for b2, b3 being Matrix of b1,b1,REAL
st b2 is_less_than b3
holds b2 is_less_or_equal_with b3;
:: MATRIX10:th 45
theorem
for b1 being natural set
for b2, b3, b4 being Matrix of b1,b1,REAL
st b2 is_less_than b3 & b3 is_less_than b4
holds b2 is_less_than b4;
:: MATRIX10:th 46
theorem
for b1 being natural set
for b2, b3, b4, b5 being Matrix of b1,b1,REAL
st b2 is_less_than b3 & b4 is_less_than b5
holds b2 + b4 is_less_than b3 + b5;
:: MATRIX10:th 47
theorem
for b1 being natural set
for b2, b3, b4 being Matrix of b1,b1,REAL
st b2 is_less_than b3
holds b2 + b4 is_less_than b3 + b4;
:: MATRIX10:th 48
theorem
for b1 being natural set
for b2, b3, b4 being Matrix of b1,b1,REAL
st b2 is_less_than b3
holds b4 - b3 is_less_than b4 - b2;
:: MATRIX10:th 49
theorem
for b1 being natural set
for b2, b3 being Matrix of b1,b1,REAL holds
|:b2 + b3:| is_less_or_equal_with |:b2:| + |:b3:|;
:: MATRIX10:th 50
theorem
for b1 being natural set
for b2, b3, b4 being Matrix of b1,b1,REAL
st b2 is_less_or_equal_with b3
holds b2 - b4 is_less_or_equal_with b3 - b4;
:: MATRIX10:th 51
theorem
for b1 being natural set
for b2, b3, b4 being Matrix of b1,b1,REAL
st b2 - b3 is_less_or_equal_with b4 - b3
holds b2 is_less_or_equal_with b4;
:: MATRIX10:th 52
theorem
for b1 being natural set
for b2, b3, b4 being Matrix of b1,b1,REAL
st b2 is_less_or_equal_with b3 - b4
holds b4 is_less_or_equal_with b3 - b2;
:: MATRIX10:th 53
theorem
for b1 being natural set
for b2, b3, b4 being Matrix of b1,b1,REAL
st b2 - b3 is_less_or_equal_with b4
holds b2 - b4 is_less_or_equal_with b3;
:: MATRIX10:th 54
theorem
for b1 being natural set
for b2, b3, b4, b5 being Matrix of b1,b1,REAL
st b2 is_less_than b3 & b4 is_less_or_equal_with b5
holds b2 - b5 is_less_than b3 - b4;
:: MATRIX10:th 55
theorem
for b1 being natural set
for b2, b3, b4, b5 being Matrix of b1,b1,REAL
st b2 is_less_or_equal_with b3 & b4 is_less_than b5
holds b2 - b5 is_less_than b3 - b4;
:: MATRIX10:th 56
theorem
for b1 being natural set
for b2, b3, b4, b5 being Matrix of b1,b1,REAL
st b2 - b3 is_less_or_equal_with b4 - b5
holds b2 - b4 is_less_or_equal_with b3 - b5;
:: MATRIX10:th 57
theorem
for b1 being natural set
for b2, b3, b4, b5 being Matrix of b1,b1,REAL
st b2 - b3 is_less_or_equal_with b4 - b5
holds b5 - b3 is_less_or_equal_with b4 - b2;
:: MATRIX10:th 58
theorem
for b1 being natural set
for b2, b3, b4, b5 being Matrix of b1,b1,REAL
st b2 - b3 is_less_or_equal_with b4 - b5
holds b5 - b4 is_less_or_equal_with b3 - b2;
:: MATRIX10:th 59
theorem
for b1 being natural set
for b2, b3, b4 being Matrix of b1,b1,REAL
st b2 + b3 is_less_or_equal_with b4
holds b2 is_less_or_equal_with b4 - b3;
:: MATRIX10:th 60
theorem
for b1 being natural set
for b2, b3, b4, b5 being Matrix of b1,b1,REAL
st b2 + b3 is_less_or_equal_with b4 + b5
holds b2 - b4 is_less_or_equal_with b5 - b3;
:: MATRIX10:th 61
theorem
for b1 being natural set
for b2, b3, b4, b5 being Matrix of b1,b1,REAL
st b2 + b3 is_less_or_equal_with b4 - b5
holds b2 + b5 is_less_or_equal_with b4 - b3;
:: MATRIX10:th 62
theorem
for b1 being natural set
for b2, b3, b4, b5 being Matrix of b1,b1,REAL
st b2 - b3 is_less_or_equal_with b4 + b5
holds b2 - b5 is_less_or_equal_with b4 + b3;
:: MATRIX10:th 63
theorem
for b1 being natural set
for b2, b3 being Matrix of b1,b1,REAL
st b2 is_less_or_equal_with b3
holds - b3 is_less_or_equal_with - b2;
:: MATRIX10:th 64
theorem
for b1 being natural set
for b2, b3 being Matrix of b1,b1,REAL
st b2 is_less_or_equal_with - b3
holds b3 is_less_or_equal_with - b2;
:: MATRIX10:th 65
theorem
for b1 being natural set
for b2, b3 being Matrix of b1,b1,REAL
st - b2 is_less_or_equal_with b3
holds - b3 is_less_or_equal_with b2;
:: MATRIX10:th 66
theorem
for b1 being natural set
for b2, b3 being Matrix of b1,b1,REAL
st b2 is Positive
holds b3 is_less_than b3 + b2;
:: MATRIX10:th 67
theorem
for b1 being natural set
for b2, b3 being Matrix of b1,b1,REAL
st b2 is Negative
holds b2 + b3 is_less_than b3;
:: MATRIX10:th 68
theorem
for b1 being natural set
for b2, b3 being Matrix of b1,b1,REAL
st b2 is Nonnegative
holds b3 is_less_or_equal_with b2 + b3;
:: MATRIX10:th 69
theorem
for b1 being natural set
for b2, b3 being Matrix of b1,b1,REAL
st b2 is Nonpositive
holds b2 + b3 is_less_or_equal_with b3;
:: MATRIX10:th 70
theorem
for b1 being natural set
for b2, b3, b4 being Matrix of b1,b1,REAL
st b2 is Nonpositive & b3 is_less_or_equal_with b4
holds b3 + b2 is_less_or_equal_with b4;
:: MATRIX10:th 71
theorem
for b1 being natural set
for b2, b3, b4 being Matrix of b1,b1,REAL
st b2 is Nonpositive & b3 is_less_than b4
holds b3 + b2 is_less_than b4;
:: MATRIX10:th 72
theorem
for b1 being natural set
for b2, b3, b4 being Matrix of b1,b1,REAL
st b2 is Negative & b3 is_less_or_equal_with b4
holds b3 + b2 is_less_than b4;
:: MATRIX10:th 73
theorem
for b1 being natural set
for b2, b3, b4 being Matrix of b1,b1,REAL
st b2 is Nonnegative & b3 is_less_or_equal_with b4
holds b3 is_less_or_equal_with b2 + b4;
:: MATRIX10:th 74
theorem
for b1 being natural set
for b2, b3, b4 being Matrix of b1,b1,REAL
st b2 is Positive & b3 is_less_or_equal_with b4
holds b3 is_less_than b2 + b4;
:: MATRIX10:th 75
theorem
for b1 being natural set
for b2, b3, b4 being Matrix of b1,b1,REAL
st b2 is Nonnegative & b3 is_less_than b4
holds b3 is_less_than b2 + b4;
:: MATRIX10:th 76
theorem
for b1 being natural set
for b2, b3 being Matrix of b1,b1,REAL
st b2 is Nonnegative
holds b3 - b2 is_less_or_equal_with b3;
:: MATRIX10:th 77
theorem
for b1 being natural set
for b2, b3 being Matrix of b1,b1,REAL
st b2 is Positive
holds b3 - b2 is_less_than b3;
:: MATRIX10:th 78
theorem
for b1 being natural set
for b2, b3 being Matrix of b1,b1,REAL
st b2 is Nonpositive
holds b3 is_less_or_equal_with b3 - b2;
:: MATRIX10:th 79
theorem
for b1 being natural set
for b2, b3 being Matrix of b1,b1,REAL
st b2 is Negative
holds b3 is_less_than b3 - b2;
:: MATRIX10:th 80
theorem
for b1 being natural set
for b2, b3 being Matrix of b1,b1,REAL
st b2 is_less_or_equal_with b3
holds b3 - b2 is Nonnegative;
:: MATRIX10:th 81
theorem
for b1 being natural set
for b2, b3, b4 being Matrix of b1,b1,REAL
st b2 is Nonnegative & b3 is_less_than b4
holds b3 - b2 is_less_than b4;
:: MATRIX10:th 82
theorem
for b1 being natural set
for b2, b3, b4 being Matrix of b1,b1,REAL
st b2 is Nonpositive & b3 is_less_or_equal_with b4
holds b3 is_less_or_equal_with b4 - b2;
:: MATRIX10:th 83
theorem
for b1 being natural set
for b2, b3, b4 being Matrix of b1,b1,REAL
st b2 is Nonpositive & b3 is_less_than b4
holds b3 is_less_than b4 - b2;
:: MATRIX10:th 84
theorem
for b1 being natural set
for b2, b3, b4 being Matrix of b1,b1,REAL
st b2 is Negative & b3 is_less_or_equal_with b4
holds b3 is_less_than b4 - b2;
:: MATRIX10:th 85
theorem
for b1 being Element of REAL
for b2 being natural set
for b3, b4 being Matrix of b2,b2,REAL
st b3 is_less_than b4 & 0 < b1
holds b1 * b3 is_less_than b1 * b4;
:: MATRIX10:th 86
theorem
for b1 being Element of REAL
for b2 being natural set
for b3, b4 being Matrix of b2,b2,REAL
st b3 is_less_than b4 & 0 <= b1
holds b1 * b3 is_less_or_equal_with b1 * b4;
:: MATRIX10:th 87
theorem
for b1 being Element of REAL
for b2 being natural set
for b3, b4 being Matrix of b2,b2,REAL
st b3 is_less_than b4 & b1 < 0
holds b1 * b4 is_less_than b1 * b3;
:: MATRIX10:th 88
theorem
for b1 being Element of REAL
for b2 being natural set
for b3, b4 being Matrix of b2,b2,REAL
st b3 is_less_than b4 & b1 <= 0
holds b1 * b4 is_less_or_equal_with b1 * b3;
:: MATRIX10:th 89
theorem
for b1 being Element of REAL
for b2 being natural set
for b3, b4 being Matrix of b2,b2,REAL
st b3 is_less_or_equal_with b4 & 0 <= b1
holds b1 * b3 is_less_or_equal_with b1 * b4;
:: MATRIX10:th 90
theorem
for b1 being Element of REAL
for b2 being natural set
for b3, b4 being Matrix of b2,b2,REAL
st b3 is_less_or_equal_with b4 & b1 <= 0
holds b1 * b4 is_less_or_equal_with b1 * b3;
:: MATRIX10:th 91
theorem
for b1, b2 being Element of REAL
for b3 being natural set
for b4, b5 being Matrix of b3,b3,REAL
st 0 <= b1 & b1 <= b2 & b4 is Nonnegative & b4 is_less_or_equal_with b5
holds b1 * b4 is_less_or_equal_with b2 * b5;
:: MATRIX10:th 92
theorem
for b1, b2 being Element of REAL
for b3 being natural set
for b4, b5 being Matrix of b3,b3,REAL
st b1 <= 0 & b2 <= b1 & b4 is Nonpositive & b5 is_less_or_equal_with b4
holds b1 * b4 is_less_or_equal_with b2 * b5;
:: MATRIX10:th 93
theorem
for b1, b2 being Element of REAL
for b3 being natural set
for b4, b5 being Matrix of b3,b3,REAL
st b1 < 0 & b2 <= b1 & b4 is Negative & b5 is_less_than b4
holds b1 * b4 is_less_than b2 * b5;
:: MATRIX10:th 94
theorem
for b1, b2 being Element of REAL
for b3 being natural set
for b4, b5 being Matrix of b3,b3,REAL
st 0 <= b1 & b1 < b2 & b4 is Nonnegative & b4 is_less_than b5
holds b1 * b4 is_less_than b2 * b5;
:: MATRIX10:th 95
theorem
for b1, b2 being Element of REAL
for b3 being natural set
for b4, b5 being Matrix of b3,b3,REAL
st 0 <= b1 & b1 < b2 & b4 is Positive & b4 is_less_or_equal_with b5
holds b1 * b4 is_less_than b2 * b5;
:: MATRIX10:th 96
theorem
for b1, b2 being Element of REAL
for b3 being natural set
for b4, b5 being Matrix of b3,b3,REAL
st 0 < b1 & b1 <= b2 & b4 is Positive & b4 is_less_than b5
holds b1 * b4 is_less_than b2 * b5;