reserve T for non empty TopSpace;
reserve A for Subset of T;

theorem
  for X being set st X in { Cl KurExSet, Cl Int KurExSet, Cl Int Cl
  KurExSet } holds X <> REAL
proof
  let X be set;
  assume
A1: X in { Cl KurExSet, Cl Int KurExSet, Cl Int Cl KurExSet };
  per cases by A1,ENUMSET1:def 1;
  suppose
A2: X = Cl KurExSet;
A3: 0 in REAL by XREAL_0:def 1;
    ( not 0 in {1})& not 0 in [. 2,+infty .[ by XXREAL_1:236, TARSKI:def 1;
    hence thesis by A2,Th10,XBOOLE_0:def 3,A3;
  end;
  suppose
    X = Cl Int KurExSet;
    hence thesis by Th24,BORSUK_5:46;
  end;
  suppose
    X = Cl Int Cl KurExSet;
    hence thesis by Th27,BORSUK_5:46;
  end;
end;
