
theorem Th44:
  for X being Subset of I(01), a being Point of I[01] st X = ]. 0,
  a .] holds X is closed
proof
  let X be Subset of I(01), a be Point of I[01];
  assume
A1: X = ]. 0, a .];
  per cases;
  suppose
A2: 0 < a;
    [#] I(01) = ]. 0, 1 .[ by Def1;
    then
A3: [#] I(01) \ X = ]. a, 1 .[ by A1,A2,XXREAL_1:187;
    1 in the carrier of I[01] by BORSUK_1:43;
    then [#] I(01) \ X is open by A3,Th43;
    hence thesis by PRE_TOPC:def 3;
  end;
  suppose
    0 >= a;
    then X = {}I(01) by A1,XXREAL_1:26;
    hence thesis;
  end;
end;
