reserve x, y, z for Real;

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