let n be Nat; for M1, M2 being Matrix of n,REAL st M1 is_less_or_equal_with M2 holds
- M2 is_less_or_equal_with - M1
let M1, M2 be Matrix of n,REAL; ( M1 is_less_or_equal_with M2 implies - M2 is_less_or_equal_with - M1 )
A1:
Indices M1 = [:(Seg n),(Seg n):]
by MATRIX_0:24;
A2:
Indices M2 = [:(Seg n),(Seg n):]
by MATRIX_0:24;
A3:
Indices (- M2) = [:(Seg n),(Seg n):]
by MATRIX_0:24;
assume A4:
M1 is_less_or_equal_with M2
; - M2 is_less_or_equal_with - M1
for i, j being Nat st [i,j] in Indices (- M2) holds
(- M2) * (i,j) <= (- M1) * (i,j)
proof
let i,
j be
Nat;
( [i,j] in Indices (- M2) implies (- M2) * (i,j) <= (- M1) * (i,j) )
assume A5:
[i,j] in Indices (- M2)
;
(- M2) * (i,j) <= (- M1) * (i,j)
then
M1 * (
i,
j)
<= M2 * (
i,
j)
by A4, A1, A3;
then
- (M2 * (i,j)) <= - (M1 * (i,j))
by XREAL_1:24;
then
(- M2) * (
i,
j)
<= - (M1 * (i,j))
by A2, A3, A5, Th2;
hence
(- M2) * (
i,
j)
<= (- M1) * (
i,
j)
by A1, A3, A5, Th2;
verum
end;
hence
- M2 is_less_or_equal_with - M1
; verum