let n be Nat; :: thesis: for M1, M2, M3 being Matrix of n, REAL st M1 - M2 is_less_or_equal_with M3 holds
M1 - M3 is_less_or_equal_with M2
let M1, M2, M3 be Matrix of n, REAL ; :: thesis: ( M1 - M2 is_less_or_equal_with M3 implies M1 - M3 is_less_or_equal_with M2 )
assume A1: M1 - M2 is_less_or_equal_with M3 ; :: thesis: M1 - M3 is_less_or_equal_with M2
A2: Indices M1 = [:(Seg n),(Seg n):] by MATRIX_1:25;
A3: ( len M1 = len M3 & width M1 = width M3 ) by Lm1;
A4: Indices (M1 - M2) = [:(Seg n),(Seg n):] by MATRIX_1:25;
A5: Indices (M1 - M3) = [:(Seg n),(Seg n):] by MATRIX_1:25;
A6: ( len M1 = len M2 & width M1 = width M2 ) by Lm1;
for i, j being Nat st [i,j] in Indices (M1 - M3) holds
(M1 - M3) * i,j <= M2 * i,j

proof

let i, j be Nat; :: thesis: ( [i,j] in Indices (M1 - M3) implies (M1 - M3) * i,j <= M2 * i,j )
assume A7: [i,j] in Indices (M1 - M3) ; :: thesis: (M1 - M3) * i,j <= M2 * i,j
then (M1 - M2) * i,j <= M3 * i,j by A1, A4, A5, Def6;
then (M1 * i,j) - (M2 * i,j) <= M3 * i,j by A2, A5, A6, A7, Th3;
then (M1 * i,j) - (M3 * i,j) <= M2 * i,j by XREAL_1:14;
hence (M1 - M3) * i,j <= M2 * i,j by A2, A5, A3, A7, Th3; :: thesis: verum

end;

hence M1 - M3 is_less_or_equal_with M2 by Def6; :: thesis: verum