let x, y be Real; :: thesis: ( x <= y & not x is zero & not y is positive implies x is negative )
assume that
A1: x <= y and
A2: not x is zero and
A3: not y is positive and
A4: not x is negative ; :: thesis: contradiction
x >= 0 by A4, XXREAL_0:def 7;
then A5: x > 0 by A2, Lm1;
y <= 0 by A3, XXREAL_0:def 6;
hence contradiction by A1, A5, Lm2; :: thesis: verum