let n be Nat; :: thesis: for M1, M2, M3, M4 being Matrix of n,REAL st M1 - M2 is_less_or_equal_with M3 - M4 holds
M1 - M3 is_less_or_equal_with M2 - M4
let M1, M2, M3, M4 be Matrix of n,REAL; :: thesis: ( M1 - M2 is_less_or_equal_with M3 - M4 implies M1 - M3 is_less_or_equal_with M2 - M4 )
A1: Indices M1 = [:(Seg n),(Seg n):] by MATRIX_0:24;
A2: Indices (M1 - M2) = [:(Seg n),(Seg n):] by MATRIX_0:24;
A3: Indices (M1 - M3) = [:(Seg n),(Seg n):] by MATRIX_0:24;
A4: Indices M3 = [:(Seg n),(Seg n):] by MATRIX_0:24;
A5: Indices M2 = [:(Seg n),(Seg n):] by MATRIX_0:24;
A6: ( len M1 = len M2 & width M1 = width M2 ) by Lm3;
A7: ( len M1 = len M3 & width M1 = width M3 ) by Lm3;
A8: ( len M3 = len M4 & width M3 = width M4 ) by Lm3;
assume A9: M1 - M2 is_less_or_equal_with M3 - M4 ; :: thesis: M1 - M3 is_less_or_equal_with M2 - M4
for i, j being Nat st [i,j] in Indices (M1 - M3) holds
(M1 - M3) * (i,j) <= (M2 - M4) * (i,j)

proof

let i, j be Nat; :: thesis: ( [i,j] in Indices (M1 - M3) implies (M1 - M3) * (i,j) <= (M2 - M4) * (i,j) )
assume A10: [i,j] in Indices (M1 - M3) ; :: thesis: (M1 - M3) * (i,j) <= (M2 - M4) * (i,j)
then (M1 - M2) * (i,j) <= (M3 - M4) * (i,j) by A9, A2, A3;
then (M1 * (i,j)) - (M2 * (i,j)) <= (M3 - M4) * (i,j) by A1, A3, A6, A10, Th3;
then (M1 * (i,j)) - (M2 * (i,j)) <= (M3 * (i,j)) - (M4 * (i,j)) by A4, A3, A8, A10, Th3;
then (M1 * (i,j)) - (M3 * (i,j)) <= (M2 * (i,j)) - (M4 * (i,j)) by XREAL_1:16;
then (M1 - M3) * (i,j) <= (M2 * (i,j)) - (M4 * (i,j)) by A1, A3, A7, A10, Th3;
hence (M1 - M3) * (i,j) <= (M2 - M4) * (i,j) by A5, A3, A6, A8, A7, A10, Th3; :: thesis: verum

end;

hence M1 - M3 is_less_or_equal_with M2 - M4 ; :: thesis: verum