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 )
A1: Indices M1 = [:(Seg n),(Seg n):] by MATRIX_0:24;
A2: Indices M2 = [:(Seg n),(Seg n):] by MATRIX_0:24;
A3: Indices (- M2) = [:(Seg n),(Seg n):] by MATRIX_0:24;
assume A4: M1 is_less_or_equal_with M2 ; :: thesis: - M2 is_less_or_equal_with - M1
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 A5: [i,j] in Indices (- M2) ; :: thesis: (- M2) * (i,j) <= (- M1) * (i,j)
then M1 * (i,j) <= M2 * (i,j) by A4, A1, A3;
then - (M2 * (i,j)) <= - (M1 * (i,j)) by XREAL_1:24;
then (- M2) * (i,j) <= - (M1 * (i,j)) by A2, A3, A5, Th2;
hence (- M2) * (i,j) <= (- M1) * (i,j) by A1, A3, A5, Th2; :: thesis: verum

end;

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