reserve r1,r2,r3 for non negative Real;
reserve n,m1 for Nat;
reserve s for Real;
reserve cn,cd,i1,j1 for Integer;
reserve r for irrational Real;
reserve q for Rational;

theorem Th6:
  TRANQN.:HWZSet(r) = HWZSet1(r)
  proof
    thus TRANQN.:HWZSet(r) c= HWZSet1(r)
    proof
      let y be object;
      assume
A2:   y in TRANQN.:HWZSet(r);
      consider p be object such that
A3:   p in dom TRANQN and
A4:   p in HWZSet(r) and
A5:   y = TRANQN.p by A2,FUNCT_1:def 6;
      consider q be Element of RAT such that
A6:   q = p by A3;
      y = denominator(q) by Def3A,A5,A6;
      hence thesis by A4,A6;
    end;
      let y be object;
      assume y in HWZSet1(r); then
      consider y1 be Nat such that
A11:  y1 = y and
A12:  ex p be Rational st p in HWZSet(r) & y1 = denominator(p);
      consider p be Rational such that
A13:  p in HWZSet(r) & y1 = denominator(p) by A12;
A14:  dom TRANQN = RAT by FUNCT_2:def 1;
      y1 = TRANQN.p by Def3A,A13;
      hence thesis by A11,A13,A14,FUNCT_1:def 6;
  end;
