let x, y be Real; :: thesis: ( x <= y & not y is zero & not x is negative implies y is positive )

assume that

A1: x <= y and

A2: not y is zero and

A3: not x is negative and

A4: not y is positive ; :: thesis: contradiction

y <= 0 by A4, XXREAL_0:def 6;

then A5: y < 0 by A2, Lm1;

x >= 0 by A3, XXREAL_0:def 7;

hence contradiction by A1, A5, Lm2; :: thesis: verum

assume that

A1: x <= y and

A2: not y is zero and

A3: not x is negative and

A4: not y is positive ; :: thesis: contradiction

y <= 0 by A4, XXREAL_0:def 6;

then A5: y < 0 by A2, Lm1;

x >= 0 by A3, XXREAL_0:def 7;

hence contradiction by A1, A5, Lm2; :: thesis: verum