let n be Nat; for M1, M2 being Matrix of n,REAL st - M2 is_less_or_equal_with M1 holds
- M1 is_less_or_equal_with M2
let M1, M2 be Matrix of n,REAL; ( - M2 is_less_or_equal_with M1 implies - M1 is_less_or_equal_with M2 )
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;
A4:
Indices (- M1) = [:(Seg n),(Seg n):]
by MATRIX_0:24;
assume A5:
- M2 is_less_or_equal_with M1
; - M1 is_less_or_equal_with M2
for i, j being Nat st [i,j] in Indices (- M1) holds
(- M1) * (i,j) <= M2 * (i,j)
proof
let i,
j be
Nat;
( [i,j] in Indices (- M1) implies (- M1) * (i,j) <= M2 * (i,j) )
assume A6:
[i,j] in Indices (- M1)
;
(- M1) * (i,j) <= M2 * (i,j)
then
(- M2) * (
i,
j)
<= M1 * (
i,
j)
by A5, A4, A3;
then
- (M2 * (i,j)) <= M1 * (
i,
j)
by A2, A4, A6, Th2;
then
- (M1 * (i,j)) <= M2 * (
i,
j)
by XREAL_1:26;
hence
(- M1) * (
i,
j)
<= M2 * (
i,
j)
by A1, A4, A6, Th2;
verum
end;
hence
- M1 is_less_or_equal_with M2
; verum