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 Th65:
  a>0 & a<1 & p>0 implies a #Q p < 1
proof
  reconsider q=0 as Rational;
  assume that
A1: a>0 and
A2: a<1 and
A3: p>0;
  1/a>1 by A1,A2,Lm4,XREAL_1:88;
  then (1/a) #Q p > (1/a) #Q q by A3,Th64;
  then (1/a) #Q p > 1 by Th47;
  then 1/a #Q p > 1 by A1,Th57;
  hence thesis by XREAL_1:185;
end;
