let n be Nat; :: thesis: 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 ; :: thesis: ( M1 is Positive & M2 is Negative implies M2 - M1 is Negative )
assume A1: ( M1 is Positive & M2 is Negative ) ; :: thesis: M2 - M1 is Negative
A2: ( Indices M1 = [:(Seg n),(Seg n):] & Indices M2 = [:(Seg n),(Seg n):] & Indices (M2 - M1) = [:(Seg n),(Seg n):] ) by MATRIX_1:25;
A3: ( len M1 = len M2 & width M1 = width M2 ) by Lm1;
for i, j being Nat st [i,j] in Indices (M2 - M1) holds
(M2 - M1) * i,j < 0

proof

let i, j be Nat; :: thesis: ( [i,j] in Indices (M2 - M1) implies (M2 - M1) * i,j < 0 )
assume A4: [i,j] in Indices (M2 - M1) ; :: thesis: (M2 - M1) * i,j < 0
then A5: M1 * i,j > 0 by A1, A2, Def1;
M2 * i,j < 0 by A1, A2, A4, Def2;
then (M1 * i,j) - (M2 * i,j) > 0 - 0 by A5;
then (- 1) * ((M1 * i,j) - (M2 * i,j)) < 0 * (- 1) by XREAL_1:71;
then (M2 * i,j) - (M1 * i,j) < 0 ;
hence (M2 - M1) * i,j < 0 by A2, A3, A4, Th3; :: thesis: verum

end;

hence M2 - M1 is Negative by Def2; :: thesis: verum