reserve Omega for non empty set;
reserve Sigma for SigmaField of Omega;
reserve S for non empty Subset of REAL;
reserve r for Real;
reserve T for Nat;

theorem PP:
  for r being Real st r>0 ex n being Nat st 1/n < r & n > 0 ::: by FRECHET:36;
  proof
    let r be Real;
    assume A1: r > 0;
    consider n being Nat such that
A2: 1 / r < n by SEQ_4:3;
    take n;
SS: r" = 1 / r & n" = 1 / n by XCMPLX_1:215;
    (r")" > n" by A2,SS,A1,XREAL_1:88;
    hence 1/n < r by XCMPLX_1:215;
    thus thesis by A1,A2;
  end;
