let n be Nat; :: thesis: for M1, M2 being Matrix of n,REAL st M1 is Negative & M2 is Negative holds
M1 + M2 is Negative
let M1, M2 be Matrix of n,REAL; :: thesis: ( M1 is Negative & M2 is Negative implies M1 + M2 is Negative )
assume that
A1: M1 is Negative and
A2: M2 is Negative ; :: thesis: M1 + M2 is Negative
A3: Indices M1 = [:(Seg n),(Seg n):] by MATRIX_0:24;
A4: Indices (M1 + M2) = [:(Seg n),(Seg n):] by MATRIX_0:24;
A5: Indices M2 = [:(Seg n),(Seg n):] by MATRIX_0:24;
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 A6: [i,j] in Indices (M1 + M2) ; :: thesis: (M1 + M2) * (i,j) < 0
then M1 * (i,j) < 0 by A1, A3, A4;
then (M1 * (i,j)) + (M2 * (i,j)) < M2 * (i,j) by XREAL_1:30;
then (M1 * (i,j)) + (M2 * (i,j)) < 0 by A2, A5, A4, A6;
hence (M1 + M2) * (i,j) < 0 by A3, A4, A6, MATRIXR1:25; :: thesis: verum

end;

hence M1 + M2 is Negative ; :: thesis: verum