let n be Nat; :: thesis: for M1, M2 being Matrix of n, REAL st M1 is Positive & M2 is Negative holds
M1 - M2 is Positive
let M1, M2 be Matrix of n, REAL ; :: thesis: ( M1 is Positive & M2 is Negative implies M1 - M2 is Positive )
assume A1: ( M1 is Positive & M2 is Negative ) ; :: thesis: M1 - M2 is Positive
A2: ( Indices M1 = [:(Seg n),(Seg n):] & Indices M2 = [:(Seg n),(Seg n):] & Indices (M1 - M2) = [:(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 (M1 - M2) holds
(M1 - M2) * i,j > 0

proof

let i, j be Nat; :: thesis: ( [i,j] in Indices (M1 - M2) implies (M1 - M2) * i,j > 0 )
assume A4: [i,j] in Indices (M1 - M2) ; :: thesis: (M1 - M2) * 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;
hence (M1 - M2) * i,j > 0 by A2, A3, A4, Th3; :: thesis: verum

end;

hence M1 - M2 is Positive by Def1; :: thesis: verum