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 Th60:
  a>=1 & p >= 0 implies a #Q p >= 1
proof
  assume that
A1: a>=1 and
A2: p>=0;
  numerator(p)>=0 by A2,RAT_1:37;
  then reconsider n = numerator(p) as Element of NAT by INT_1:3;
A3: a #Z numerator(p) = a |^ n by Th36;
  a |^ n >= 1 |^ n by A1,Th9;
  then
A4: a #Z numerator(p) >= 1 by A3;
  denominator(p) >= 1 by RAT_1:11;
  hence thesis by A4,Th29;
end;
