reserve n   for Nat,
        r,s for Real,
        x,y for Element of REAL n,
        p,q for Point of TOP-REAL n,
        e   for Point of Euclid n;

theorem Th20:
  for O being non empty open Subset of TOP-REAL n holds
  ex p being Element of RAT n st p in O
  proof
    let O be non empty open Subset of TOP-REAL n;
    RAT n is dense Subset of TOP-REAL n by Th15;
    then O meets RAT n by TOPS_1:45;
    then consider x be object such that
A1: x in O and
A2: x in RAT n by XBOOLE_0:3;
    reconsider x as Element of RAT n by A2;
    take x;
    thus thesis by A1;
  end;
