let x, y be real number ; :: thesis: ( x <= y & not x is zero & not y is positive implies x is negative )
assume A1: ( x <= y & not x is zero & not y is positive & not x is negative ) ; :: thesis: contradiction
then ( x >= 0 & y <= 0 ) by XXREAL_0:def 6, XXREAL_0:def 7;
then ( x > 0 & y <= 0 ) by A1, Lm1;
hence contradiction by A1, Lm2; :: thesis: verum