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