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
  a>0 & a<=1 & p<=0 implies a #Q p >= 1
proof
  assume that
A1: a>0 and
A2: a<=1 and
A3: p<=0;
  1/a >= 1 by A1,A2,Lm4,XREAL_1:85;
  then (1/a) #Q p <= 1 by A3,Th61;
  then
A4: 1/a #Q p <= 1 by A1,Th57;
  a #Q p > 0 by A1,Th52;
  hence thesis by A4,XREAL_1:187;
end;
