let n be Nat; :: thesis: for M1, M2, M3 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, M2, M3 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:
A2: Indices M1 = [:(Seg n),(Seg n):] by MATRIX_0:24;
A3: width M2 = width M3 by Lm3;
A4: ( Indices M2 = [:(Seg n),(Seg n):] & len M2 = len M3 ) by ;
A5: Indices (M1 - M3) = [:(Seg n),(Seg n):] by MATRIX_0:24;
A6: ( len M1 = len M3 & width M1 = width M3 ) by Lm3;
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;
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_1:6;
hence M1 * (i,j) <= M2 * (i,j) ; :: thesis: verum
end;
hence M1 is_less_or_equal_with M2 ; :: thesis: verum