let n be Nat; :: thesis: for M1, M2 being Matrix of n, REAL st M1 is_less_or_equal_with - M2 holds
M1 + M2 is Nonpositive
let M1, M2 be Matrix of n, REAL ; :: thesis: ( M1 is_less_or_equal_with - M2 implies M1 + M2 is Nonpositive )
assume A1: M1 is_less_or_equal_with - M2 ; :: thesis: M1 + M2 is Nonpositive
A2: ( Indices M1 = [:(Seg n),(Seg n):] & Indices M2 = [:(Seg n),(Seg n):] & Indices (- M2) = [:(Seg n),(Seg n):] & Indices (M1 + M2) = [:(Seg n),(Seg n):] ) by MATRIX_1:25;
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 A3: [i,j] in Indices (M1 + M2) ; :: thesis: (M1 + M2) * i,j <= 0
then M1 * i,j <= (- M2) * i,j by A1, A2, Def6;
then M1 * i,j <= - (M2 * i,j) by A2, A3, Th2;
then (M1 * i,j) + (M2 * i,j) <= 0 by XREAL_1:61;
hence (M1 + M2) * i,j <= 0 by A2, A3, MATRIXR1:25; :: thesis: verum

end;

hence M1 + M2 is Nonpositive by Def3; :: thesis: verum