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

proof

let i, j be Nat; :: thesis: ( [i,j] in Indices M1 implies M1 * i,j <= M2 * i,j )
assume A7: [i,j] in Indices M1 ; :: thesis: M1 * i,j <= M2 * i,j
then (M1 - M3) * i,j <= (M2 - M3) * i,j by A1, A2, A5, Def6;
then (M1 * i,j) - (M3 * i,j) <= (M2 - M3) * i,j by A6, A7, Th3;
then (M1 * i,j) - (M3 * i,j) <= (M2 * i,j) - (M3 * i,j) by A2, A4, A3, A7, Th3;
then ((M1 * i,j) - (M3 * i,j)) + (M3 * i,j) <= ((M2 * i,j) - (M3 * i,j)) + (M3 * i,j) by XREAL_1:8;
hence M1 * i,j <= M2 * i,j ; :: thesis: verum

end;

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