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

proof

let i, j be Nat; :: thesis: ( [i,j] in Indices M3 implies M3 * (i,j) <= (M2 - M1) * (i,j) )
assume A7: [i,j] in Indices M3 ; :: thesis: M3 * (i,j) <= (M2 - M1) * (i,j)
then M1 * (i,j) <= (M2 - M3) * (i,j) by A1, A2, A3;
then M1 * (i,j) <= (M2 * (i,j)) - (M3 * (i,j)) by A5, A3, A4, A7, Th3;
then M3 * (i,j) <= (M2 * (i,j)) - (M1 * (i,j)) by XREAL_1:11;
hence M3 * (i,j) <= (M2 - M1) * (i,j) by A5, A3, A6, A7, Th3; :: thesis: verum

end;

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