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):] 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

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):] 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

hence
M2 - M1 is Negative
; :: thesis: verum
let i, j be Nat; :: thesis: ( [i,j] in Indices (M2 - M1) implies (M2 - M1) * (i,j) < 0 )

assume A5: [i,j] in Indices (M2 - M1) ; :: thesis: (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; :: thesis: verum

end;assume A5: [i,j] in Indices (M2 - M1) ; :: thesis: (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; :: thesis: verum