
theorem Th45:
  for X being Subset of I(01), a being Point of I[01] st X = [. a,
  1 .[ holds X is closed
proof
A1: 0 in the carrier of I[01] by BORSUK_1:43;
  let X be Subset of I(01), a be Point of I[01];
  assume
A2: X = [. a, 1 .[;
  per cases;
  suppose
A3: X is non empty;
A4: a <= 1 by BORSUK_1:43;
    a <> 1 by A2,A3,XXREAL_1:18;
    then
A5: a < 1 by A4,XXREAL_0:1;
    [#] I(01) = ]. 0, 1 .[ by Def1;
    then [#] I(01) \ X = ]. 0, a .[ by A2,A5,XXREAL_1:195;
    then [#] I(01) \ X is open by A1,Th43;
    hence thesis by PRE_TOPC:def 3;
  end;
  suppose
    X is empty;
    hence thesis;
  end;
end;
