let n be Nat; for M1, M2 being Matrix of n,REAL st M1 is_less_than M2 holds
M1 - M2 is Negative
let M1, M2 be Matrix of n,REAL; ( M1 is_less_than M2 implies M1 - M2 is Negative )
assume A1:
M1 is_less_than M2
; M1 - M2 is Negative
A2:
( Indices M1 = [:(Seg n),(Seg n):] & Indices (M1 - M2) = [:(Seg n),(Seg n):] )
by MATRIX_0:24;
A3:
( len M1 = len M2 & width M1 = width M2 )
by Lm3;
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 A4:
[i,j] in Indices (M1 - M2)
;
(M1 - M2) * (i,j) < 0
then
M1 * (
i,
j)
< M2 * (
i,
j)
by A1, A2;
then
(M1 * (i,j)) - (M2 * (i,j)) < 0
by XREAL_1:49;
hence
(M1 - M2) * (
i,
j)
< 0
by A2, A3, A4, Th3;
verum
end;
hence
M1 - M2 is Negative
; verum