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

theorem
  Sorgenfrey-line is Tychonoff
proof
  set X = Sorgenfrey-line;
A1: the carrier of X = REAL by TOPGEN_3:def 2;
  consider B being Subset-Family of REAL such that
A2: the topology of X = UniCl B and
A3: B = {[.x,q.[ where x,q is Real: x < q & q is rational}
    by TOPGEN_3:def 2;
  B c= UniCl B by CANTOR_1:1;
  then B is Basis of X by A1,A2,CANTOR_1:def 2,TOPS_2:64;
  then reconsider B as prebasis of X by YELLOW_9:27;
  now
    let x be Point of X;
    let V be Subset of X;
    assume that
A4: x in V and
A5: V in B;
    consider a,q being Real such that
A6: V = [.a,q.[ and
    a < q and
A7: q is rational by A5,A3;
    consider f being continuous Function of X, I[01] such that
A8: for b being Point of Sorgenfrey-line holds (b in [.a,q.[ implies
    f.b = 0) & (not b in [.a,q.[ implies f.b = 1) by A7,Th58;
    take f;
    thus f.x = 0 by A4,A6,A8;
    thus f.:V` c= {1}
    proof
      let u be object;
      assume u in f.:V`;
      then consider b being Point of X such that
A9:   b in V` and
A10:  u = f.b by FUNCT_2:65;
      not b in [.a,q.[ by A6,A9,XBOOLE_0:def 5;
      then u = 1 by A8,A10;
      hence thesis by TARSKI:def 1;
    end;
  end;
  hence thesis by Th52,Th53;
end;
