let n be Nat; for M1, M2 being Matrix of n,REAL st M1 is Positive holds
M2 - M1 is_less_than M2
let M1, M2 be Matrix of n,REAL; ( M1 is Positive implies M2 - M1 is_less_than M2 )
assume A1:
M1 is Positive
; M2 - M1 is_less_than M2
A2:
Indices M1 = [:(Seg n),(Seg n):]
by MATRIX_0:24;
A3:
width M1 = width M2
by Lm3;
A4:
Indices (M2 - M1) = [:(Seg n),(Seg n):]
by MATRIX_0:24;
A5:
( Indices M2 = [:(Seg n),(Seg n):] & len M1 = len M2 )
by Lm3, MATRIX_0:24;
for i, j being Nat st [i,j] in Indices (M2 - M1) holds
(M2 - M1) * (i,j) < M2 * (i,j)
proof
let i,
j be
Nat;
( [i,j] in Indices (M2 - M1) implies (M2 - M1) * (i,j) < M2 * (i,j) )
assume A6:
[i,j] in Indices (M2 - M1)
;
(M2 - M1) * (i,j) < M2 * (i,j)
then
M1 * (
i,
j)
> 0
by A1, A2, A4;
then
(M2 * (i,j)) - (M1 * (i,j)) < M2 * (
i,
j)
by XREAL_1:44;
hence
(M2 - M1) * (
i,
j)
< M2 * (
i,
j)
by A4, A5, A3, A6, Th3;
verum
end;
hence
M2 - M1 is_less_than M2
; verum