reserve x for object,
  a,b for Real,
  k,k1,i1,j1,w for Nat,
  m,m1,n,n1 for Integer;
reserve p,q for Rational;

theorem Th6:
  ex m,k st k <> 0 & p = m/k & for n,w st w <> 0 & p = n/w holds k <= w
proof
  defpred P[Nat] means ($1<>0 & ex m1 st p=m1/$1);
  ex m,k st k>0 & p=m/k by Th5;
  then
A1: ex l be Nat st P[l];
  ex k1 be Nat st P[k1] & for l1 be Nat st P[l1] holds k1<=l1 from NAT_1:
  sch 5(A1);
  then consider k1 be Nat such that
A2: k1<>0 and
A3: ex m1 st p=m1/k1 and
A4: for l1 be Nat st (l1<>0 & ex n1 st p=n1/l1) holds k1<=l1;
  reconsider k1 as Element of NAT by ORDINAL1:def 12;
  consider m1 such that
A5: p=m1/k1 and
A6: for l1 be Nat st (l1<>0 & ex n1 st p=n1/l1) holds k1<=l1 by A3,A4;
  take m1, k1;
  thus k1<>0 by A2;
  thus m1/k1=p by A5;
  thus thesis by A6;
end;
