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 Th54:
  a>0 implies 1 / a #Q p = a #Q (-p)
proof
  assume a>0;
  then
A1: a #Z numerator(p) > 0 by Th39;
  thus a #Q (-p) = (denominator(-p)) -Root (a #Z (-numerator(p))) by RAT_1:43
    .= (denominator(p)) -Root (a #Z (-numerator(p))) by RAT_1:43
    .= (denominator(p)) -Root (1/a #Z numerator(p)) by Th41
    .= 1 / a #Q p by A1,Th23,RAT_1:11;
end;
