reserve A, B, X, Y for set;
reserve R, S, T for non empty TopSpace;

theorem
  T is Hausdorff iff for A being Subset of [:T,T:] st A = id the carrier
  of T holds A is closed
proof
  reconsider f = pr1(the carrier of T,the carrier of T), g = pr2(the carrier
  of T,the carrier of T) as continuous Function of [:T,T:],T by Th39,Th40;
  reconsider A = id the carrier of T as Subset of [:T,T:] by BORSUK_1:def 2;
  hereby
    assume
A1: T is Hausdorff;
    let A be Subset of [:T,T:];
    assume A = id the carrier of T;
    then A = {p where p is Point of [:T,T:]: f.p = g.p} by Th37;
    hence A is closed by A1,Th45;
  end;
  assume
  for A being Subset of [:T,T:] st A = id the carrier of T holds A is closed;
  then [#][:T,T:] = [:[#]T,[#]T:] & A is closed by BORSUK_1:def 2;
  then [:[#]T,[#]T:] \ A is open;
  then consider SF being Subset-Family of [:T,T:] such that
A2: [:[#]T,[#]T:] \ A = union SF and
A3: for e being set st e in SF ex X1, Y1 being Subset of T st e = [:X1,
  Y1:] & X1 is open & Y1 is open by BORSUK_1:5;
  let p, q be Point of T;
  assume not p = q;
  then
  the carrier of [:T,T:] = [:the carrier of T,the carrier of T:] & not [p
  ,q] in id [#]T by BORSUK_1:def 2,RELAT_1:def 10;
  then [p,q] in [:[#]T,[#]T:] \ A by XBOOLE_0:def 5;
  then consider XY being set such that
A4: [p,q] in XY and
A5: XY in SF by A2,TARSKI:def 4;
  consider X1, Y1 being Subset of T such that
A6: XY = [:X1,Y1:] and
A7: X1 is open & Y1 is open by A3,A5;
  take X1, Y1;
  thus X1 is open & Y1 is open by A7;
  thus p in X1 & q in Y1 by A4,A6,ZFMISC_1:87;
  assume X1 /\ Y1 <> {};
  then consider w being object such that
A8: w in X1 /\ Y1 by XBOOLE_0:def 1;
  w in X1 & w in Y1 by A8,XBOOLE_0:def 4;
  then [w,w] in XY by A6,ZFMISC_1:87;
  then [w,w] in union SF by A5,TARSKI:def 4;
  then not [w,w] in A by A2,XBOOLE_0:def 5;
  hence contradiction by A8,RELAT_1:def 10;
end;
