reserve x, y, z for Real;

theorem
  x <= y & x is non zero & y is non positive implies x is negative
proof
  assume that
A1: x <= y and
A2: x is non zero and
A3: y is non positive and
A4: x is non negative;
  x >= 0 by A4,XXREAL_0:def 7;
  then
A5: x > 0 by A2,Lm1;
  y <= 0 by A3,XXREAL_0:def 6;
  hence thesis by A1,A5,Lm2;
end;
