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 Th63:
  a>=1 & p>=q implies a #Q p >= a #Q q
proof
  assume that
A1: a>=1 and
A2: p>=q;
  per cases by A2,XXREAL_0:1;
  suppose
    p=q;
    hence thesis;
  end;
  suppose
    p>q;
    then
A3: p-q>=0 by XREAL_1:50;
A4: a #Q q <> 0 by A1,Th52;
A5: a #Q p / a #Q q = a #Q (p-q) by A1,Th55;
    a #Q q >= 0 by A1,Th52;
    then a #Q p / a #Q q * a #Q q >= 1 * a #Q q by A1,A3,A5,Th60,XREAL_1:64;
    hence thesis by A4,XCMPLX_1:87;
  end;
end;
