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_0:24;
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;
then M1 * (i,j) <= - (M2 * (i,j)) by A3, Th2;
then M2 * (i,j) <= - (M1 * (i,j)) by XREAL_1:25;
hence M2 * (i,j) <= (- M1) * (i,j) by A1, A3, Th2; :: thesis: verum

end;

hence M2 is_less_or_equal_with - M1 ; :: thesis: verum