reserve x, y, z for Real;

theorem
  x <= y & y is non positive implies x is non positive
proof
  assume that
A1: x <= y and
A2: y is non positive and
A3: x is positive;
  x > 0 by A3,XXREAL_0:def 6;
  then y > 0 by A1,Lm2;
  hence thesis by A2,XXREAL_0:def 6;
end;
