reserve x for set;
reserve a, b, c for Real;
reserve m, n, m1, m2 for Nat;
reserve k, l for Integer;
reserve p, q for Rational;
reserve s1, s2 for Real_Sequence;

theorem Th62:
  a>1 & p>0 implies a #Q p > 1
proof
  assume that
A1: a>1 and
A2: p>0;
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 Th36;
  then a #Z numerator(p) > 1 by A4;
  then (denominator(p)) -Root (a #Z numerator(p))>(denominator(p)) -Root 1 by
Th28,RAT_1:11;
  hence thesis by Th20,RAT_1:11;
end;
