let n be Nat; :: thesis: for M1, M2 being Matrix of n, REAL st M1 is_less_or_equal_with M2 holds
- M2 is_less_or_equal_with - M1
let M1, M2 be Matrix of n, REAL ; :: thesis: ( M1 is_less_or_equal_with M2 implies - M2 is_less_or_equal_with - M1 )
assume A1: M1 is_less_or_equal_with M2 ; :: thesis: - M2 is_less_or_equal_with - M1
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 (- M2) holds
(- M2) * i,j <= (- M1) * i,j

proof

let i, j be Nat; :: thesis: ( [i,j] in Indices (- M2) implies (- M2) * i,j <= (- M1) * i,j )
assume A3: [i,j] in Indices (- M2) ; :: thesis: (- M2) * i,j <= (- M1) * i,j
then M1 * i,j <= M2 * i,j by A1, A2, Def6;
then - (M2 * i,j) <= - (M1 * i,j) by XREAL_1:26;
then (- M2) * i,j <= - (M1 * i,j) by A2, A3, Th2;
hence (- M2) * i,j <= (- M1) * i,j by A2, A3, Th2; :: thesis: verum

end;

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