let a be real number ; :: thesis: for p being Rational st a > 1 & p > 0 holds
a #Q p > 1
let p be Rational; :: thesis: ( a > 1 & p > 0 implies a #Q p > 1 )
assume that
A1: a > 1 and
A2: p > 0 ; :: thesis: a #Q p > 1
A3: numerator p > 0 by A2, RAT_1:38;
then reconsider n = numerator p as Element of NAT by INT_1:3;
n >= 0 + 1 by A3, NAT_1:13;
then A4: a |^ n > 1 |^ n by A1, Lm1;
a #Z (numerator p) = a |^ n by Th46;
then a #Z (numerator p) > 1 by A4, NEWTON:10;
then (denominator p) -Root (a #Z (numerator p)) > (denominator p) -Root 1 by Th37, RAT_1:11;
hence a #Q p > 1 by Th29, RAT_1:11; :: thesis: verum