let n be Nat; :: thesis: for M1, M3, M2 being Matrix of n, REAL st M1 is Negative & M3 is_less_or_equal_with M2 holds
M3 + M1 is_less_than M2
let M1, M3, M2 be Matrix of n, REAL ; :: thesis: ( M1 is Negative & M3 is_less_or_equal_with M2 implies M3 + M1 is_less_than M2 )
assume A1: ( M1 is Negative & M3 is_less_or_equal_with M2 ) ; :: thesis: M3 + M1 is_less_than M2
A2: ( Indices M1 = [:(Seg n),(Seg n):] & Indices M2 = [:(Seg n),(Seg n):] & Indices M3 = [:(Seg n),(Seg n):] & Indices (M3 + M1) = [:(Seg n),(Seg n):] ) by MATRIX_1:25;
for i, j being Nat st [i,j] in Indices (M3 + M1) holds
(M3 + M1) * i,j < M2 * i,j

proof

let i, j be Nat; :: thesis: ( [i,j] in Indices (M3 + M1) implies (M3 + M1) * i,j < M2 * i,j )
assume A3: [i,j] in Indices (M3 + M1) ; :: thesis: (M3 + M1) * i,j < M2 * i,j
then A4: M1 * i,j < 0 by A1, A2, Def2;
M3 * i,j <= M2 * i,j by A1, A2, A3, Def6;
then (M3 * i,j) + (M1 * i,j) < M2 * i,j by A4, XREAL_1:39;
hence (M3 + M1) * i,j < M2 * i,j by A2, A3, MATRIXR1:25; :: thesis: verum

end;

hence M3 + M1 is_less_than M2 by Def5; :: thesis: verum