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):] & Indices M2 = [:(Seg n),(Seg n):] ) by MATRIX_1:25;
assume A2: 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 A3: [i,j] in Indices M2 ; :: thesis: M2 * i,j <= (- M1) * i,j
then M1 * i,j <= (- M2) * i,j by A2, A1, Def6;
then M1 * i,j <= - (M2 * i,j) by A3, Th2;
then M2 * i,j <= - (M1 * i,j) by XREAL_1:27;
hence M2 * i,j <= (- M1) * i,j by A1, A3, Th2; :: thesis: verum

end;

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