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;
reserve c0,c1,c2,u,a0,b0 for Real;
reserve a,b for Real;
reserve n for Integer;
reserve a1,a2,b1,b2,c1,c2 for Element of REAL;
reserve eps for positive Real;
reserve r1 for non negative Real;
reserve q,q1 for Element of RAT;

theorem Th42:
  ex q be Element of RAT st denominator(q) > [\r1/]+1 & q in HWZSet(r)
  proof
     0 < [\r1/]+1 by INT_1:29; then
     reconsider m = [\r1/]+1 as Nat;
     ex n st n in HWZSet1(r) & n > m
     proof
       assume
A1:    not ex n st n in HWZSet1(r) & n > m;
A2:    for n st n in HWZSet1(r) holds n in Seg(m)
       proof
         let n;
         assume
A3:      n in HWZSet1(r); then
         n > 0 by Th10; then
A4:      n+0 >= 1 by NAT_1:19;
         n <= m by A1,A3;
         hence thesis by A4;
       end;
       Seg(m) c= Segm(m+1) by AFINSQ_1:3; then
       HWZSet1(r)c=Segm(m+1) by A2;
       hence thesis;
     end; then
     consider n such that
A8:  n in HWZSet1(r) and
A9:  n > m;
     ex n1 be Nat st n1 = n &
   ex p be Rational st p in HWZSet(r) & n1 = denominator(p) by A8;
   hence thesis by A9;
   end;
