let a be Real; :: thesis: for p, q being Rational st a >= 1 & p >= q holds
a #Q p >= a #Q q
let p, q be Rational; :: thesis: ( a >= 1 & p >= q implies a #Q p >= a #Q q )
assume that
A1: a >= 1 and
A2: p >= q ; :: thesis: a #Q p >= a #Q q

per cases ( p = q or p > q ) by A2, XXREAL_0:1;

suppose p = q ; :: thesis: a #Q p >= a #Q q

hence a #Q p >= a #Q q ; :: thesis: verum

end;

suppose p > q ; :: thesis: a #Q p >= a #Q q

then A3: p - q >= 0 by XREAL_1:50;
A4: a #Q q <> 0 by A1, Th52;
A5: (a #Q p) / (a #Q q) = a #Q (p - q) by A1, Th55;
a #Q q >= 0 by A1, Th52;
then ((a #Q p) / (a #Q q)) * (a #Q q) >= 1 * (a #Q q) by A1, A3, A5, Th60, XREAL_1:64;
hence a #Q p >= a #Q q by A4, XCMPLX_1:87; :: thesis: verum

end;

end;