reserve x,y for Real,
  u,v,w for set,
  r for positive Real;

theorem Th43:
  RAT is dense Subset of Sorgenfrey-line
proof
  reconsider A = RAT as Subset of Sorgenfrey-line by NUMBERS:12,TOPGEN_3:def 2;
  consider B being Subset-Family of REAL such that
A1: the topology of Sorgenfrey-line = UniCl B and
A2: B = {[.x,q.[ where x,q is Real: x < q & q is rational}
          by TOPGEN_3:def 2;
  the carrier of Sorgenfrey-line = REAL by TOPGEN_3:def 2;
  then
A3: B is Basis of Sorgenfrey-line by A1,YELLOW_9:22;
  A is dense
  proof
    thus Cl A c= the carrier of Sorgenfrey-line;
    let x be object;
    assume x in the carrier of Sorgenfrey-line;
    then reconsider x as Point of Sorgenfrey-line;
    now
      let C be Subset of Sorgenfrey-line;
      assume C in B;
      then consider y,q being Real such that
A4:   C = [.y,q.[ and
A5:   y < q and
      q is rational by A2;
      assume x in C;
      consider r being Rational such that
A6:   y < r and
A7:   r < q by A5,RAT_1:7;
A8:   r in A by RAT_1:def 2;
      r in C by A4,A6,A7,XXREAL_1:3;
      hence A meets C by A8,XBOOLE_0:3;
    end;
    hence thesis by A3,YELLOW_9:37;
  end;
  hence thesis;
end;
