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

theorem Th49:
  for X being set st X in { Int KurExSet, Int Cl KurExSet, Int Cl
  Int KurExSet } holds X <> REAL
proof
  let X be set;
  assume
A1: X in { Int KurExSet, Int Cl KurExSet, Int Cl Int KurExSet };
  per cases by A1,ENUMSET1:def 1;
  suppose
A2: X = Int KurExSet;
     5 in REAL by XREAL_0:def 1;
    hence thesis by Th23,XXREAL_1:205,A2;
  end;
  suppose
    X = Int Cl KurExSet;
    hence thesis by Th26,BORSUK_5:45;
  end;
  suppose
    X = Int Cl Int KurExSet;
    hence thesis by Th25,BORSUK_5:45;
  end;
end;
