let n be Nat; :: thesis: for M2, M1 being Matrix of n, REAL st - M2 is_less_or_equal_with M1 holds
- M1 is_less_or_equal_with M2
let M2, M1 be Matrix of n, REAL ; :: thesis: ( - M2 is_less_or_equal_with M1 implies - M1 is_less_or_equal_with M2 )
assume A1: - M2 is_less_or_equal_with M1 ; :: thesis: - M1 is_less_or_equal_with M2
A2: ( Indices M1 = [:(Seg n),(Seg n):] & Indices M2 = [:(Seg n),(Seg n):] & Indices (- M1) = [:(Seg n),(Seg n):] & Indices (- M2) = [:(Seg n),(Seg n):] ) by MATRIX_1:25;
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 A3: [i,j] in Indices (- M1) ; :: thesis: (- M1) * i,j <= M2 * i,j
then (- M2) * i,j <= M1 * i,j by A1, A2, Def6;
then - (M2 * i,j) <= M1 * i,j by A2, A3, Th2;
then - (M1 * i,j) <= M2 * i,j by XREAL_1:28;
hence (- M1) * i,j <= M2 * i,j by A2, A3, Th2; :: thesis: verum

end;

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