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 Th64:
  a>1 & p>q implies a #Q p > a #Q q
proof
  assume that
A1: a>1 and
A2: p>q;
A3: p-q>0 by A2,XREAL_1:50;
A4: a #Q p / a #Q q = a #Q (p-q) by A1,Th55;
A5: a #Q q <> 0 by A1,Th52;
  a #Q q > 0 by A1,Th52;
  then a #Q p / a #Q q * a #Q q > 1 * a #Q q by A1,A3,A4,Th62,XREAL_1:68;
  hence thesis by A5,XCMPLX_1:87;
end;
