let n be Nat; for M1, M2 being Matrix of n,REAL st M2 is_less_than M1 holds
M1 - M2 is Positive
let M1, M2 be Matrix of n,REAL; ( M2 is_less_than M1 implies M1 - M2 is Positive )
assume A1:
M2 is_less_than M1
; M1 - M2 is Positive
A2:
Indices M2 = [:(Seg n),(Seg n):]
by MATRIX_0:24;
A3:
width M1 = width M2
by Lm3;
A4:
Indices (M1 - M2) = [:(Seg n),(Seg n):]
by MATRIX_0:24;
A5:
( Indices M1 = [:(Seg n),(Seg n):] & len M1 = len M2 )
by Lm3, MATRIX_0:24;
for i, j being Nat st [i,j] in Indices (M1 - M2) holds
(M1 - M2) * (i,j) > 0
proof
let i,
j be
Nat;
( [i,j] in Indices (M1 - M2) implies (M1 - M2) * (i,j) > 0 )
assume A6:
[i,j] in Indices (M1 - M2)
;
(M1 - M2) * (i,j) > 0
then
M1 * (
i,
j)
> M2 * (
i,
j)
by A1, A2, A4;
then
(M1 * (i,j)) - (M2 * (i,j)) > 0
by XREAL_1:50;
hence
(M1 - M2) * (
i,
j)
> 0
by A4, A5, A3, A6, Th3;
verum
end;
hence
M1 - M2 is Positive
; verum