for i, j being Nat st [i,j] in Indices (M @) holds
(M @) * (i,j) <= 0

proof

let i, j be Nat; :: thesis: ( [i,j] in Indices (M @) implies (M @) * (i,j) <= 0 )
assume [i,j] in Indices (M @) ; :: thesis: (M @) * (i,j) <= 0
then A1: [j,i] in Indices M by MATRIX_0:def 6;
then (M @) * (i,j) = M * (j,i) by MATRIX_0:def 6;
hence (M @) * (i,j) <= 0 by A1, Def3; :: thesis: verum

end;

hence M @ is Nonpositive ; :: thesis: verum