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

proof

let i, j be Nat; :: thesis: ( [i,j] in Indices (M2 - M1) implies (M2 - M1) * i,j <= M2 * i,j )
assume A4: [i,j] in Indices (M2 - M1) ; :: thesis: (M2 - M1) * i,j <= M2 * i,j
then M1 * i,j >= 0 by A1, A2, Def4;
then (M2 * i,j) - (M1 * i,j) <= M2 * i,j by XREAL_1:45;
hence (M2 - M1) * i,j <= M2 * i,j by A2, A3, A4, Th3; :: thesis: verum

end;

hence M2 - M1 is_less_or_equal_with M2 by Def6; :: thesis: verum