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 M4 - M2
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 M4 - M2 )
assume A1: M1 + M2 is_less_or_equal_with M3 + M4 ; :: thesis: M1 - M3 is_less_or_equal_with M4 - M2
A2: Indices M1 = [:(Seg n),(Seg n):] by MATRIX_1:25;
A3: width M2 = width M4 by Lm1;
A4: Indices (M1 + M2) = [:(Seg n),(Seg n):] by MATRIX_1:25;
A5: ( Indices M4 = [:(Seg n),(Seg n):] & len M2 = len M4 ) by Lm1, MATRIX_1:25;
A6: Indices M3 = [:(Seg n),(Seg n):] by MATRIX_1:25;
A7: Indices (M1 - M3) = [:(Seg n),(Seg n):] by MATRIX_1:25;
A8: ( len M1 = len M3 & width M1 = width M3 ) by Lm1;
for i, j being Nat st [i,j] in Indices (M1 - M3) holds
(M1 - M3) * i,j <= (M4 - M2) * i,j

proof

let i, j be Nat; :: thesis: ( [i,j] in Indices (M1 - M3) implies (M1 - M3) * i,j <= (M4 - M2) * i,j )
assume A9: [i,j] in Indices (M1 - M3) ; :: thesis: (M1 - M3) * i,j <= (M4 - M2) * i,j
then (M1 + M2) * i,j <= (M3 + M4) * i,j by A1, A4, A7, Def6;
then (M1 * i,j) + (M2 * i,j) <= (M3 + M4) * i,j by A2, A7, A9, MATRIXR1:25;
then (M1 * i,j) + (M2 * i,j) <= (M3 * i,j) + (M4 * i,j) by A6, A7, A9, MATRIXR1:25;
then (M1 * i,j) - (M3 * i,j) <= (M4 * i,j) - (M2 * i,j) by XREAL_1:23;
then (M1 - M3) * i,j <= (M4 * i,j) - (M2 * i,j) by A2, A7, A8, A9, Th3;
hence (M1 - M3) * i,j <= (M4 - M2) * i,j by A7, A5, A3, A9, Th3; :: thesis: verum

end;

hence M1 - M3 is_less_or_equal_with M4 - M2 by Def6; :: thesis: verum