let a be Real; :: thesis: for p being Rational st a > 0 & a < 1 & p > 0 holds
a #Q p < 1
let p be Rational; :: thesis: ( a > 0 & a < 1 & p > 0 implies a #Q p < 1 )
reconsider q = 0 as Rational ;
assume that
A1: a > 0 and
A2: a < 1 and
A3: p > 0 ; :: thesis: a #Q p < 1
1 / a > 1 by A1, A2, Lm4, XREAL_1:88;
then (1 / a) #Q p > (1 / a) #Q q by A3, Th64;
then (1 / a) #Q p > 1 by Th47;
then 1 / (a #Q p) > 1 by A1, Th57;
hence a #Q p < 1 by XREAL_1:185; :: thesis: verum