reserve x,y for set;
reserve s,s1,s2,s4,r,r1,r2 for Real;
reserve n,m,i,j for Element of NAT;

theorem Th1:
  for r st r>0 ex n being Nat st n>0 & 1/n < r
proof
  let r such that
A1: r>0;
A2: 1/r>0 by A1;
  consider n being Nat such that
A3: 1/r < n by SEQ_4:3;
  1/r*r<n*r by A1,A3,XREAL_1:68;
  then 1<n*r by A1,XCMPLX_1:87;
  then 1/n<r by A3,A2,XREAL_1:83;
  hence thesis by A3,A2;
end;
