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

theorem Th53:
  Sorgenfrey-line is T_1
proof
  set T = Sorgenfrey-line;
  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;
  let x,y be Point of T;
  reconsider a = x, b = y as Element of REAL by TOPGEN_3:def 2;
A3: B c= the topology of T by A1,CANTOR_1:1;
  assume
A4: x <> y;
  per cases by A4,XXREAL_0:1;
  suppose
A5: a < b;
    b < b+1 by XREAL_1:29;
    then consider q being Rational such that
A6: b < q and
    q < b+1 by RAT_1:7;
    [.b,q.[ in B by A2,A6;
    then
A7: [.b,q.[ in the topology of T by A3;
    consider w being Rational such that
A8: a < w and
A9: w < b by A5,RAT_1:7;
    [.a,w.[ in B by A2,A8;
    then [.a,w.[ in the topology of T by A3;
    then reconsider U = [.a,w.[, V = [.b,q.[ as open Subset of T by A7,
PRE_TOPC:def 2;
    take U,V;
    thus U is open & V is open;
    thus thesis by A5,A8,A9,A6,XXREAL_1:3;
  end;
  suppose
A10: a > b;
    a < a+1 by XREAL_1:29;
    then consider q being Rational such that
A11: a < q and
    q < a+1 by RAT_1:7;
    [.a,q.[ in B by A2,A11;
    then
A12: [.a,q.[ in the topology of T by A3;
    consider w being Rational such that
A13: b < w and
A14: w < a by A10,RAT_1:7;
    [.b,w.[ in B by A2,A13;
    then [.b,w.[ in the topology of T by A3;
    then reconsider V = [.b,w.[, U = [.a,q.[ as open Subset of T by A12,
PRE_TOPC:def 2;
    take U,V;
    thus U is open & V is open;
    thus thesis by A10,A13,A14,A11,XXREAL_1:3;
  end;
end;
