
theorem Th42:
  for X being Subset of I[01], a, b being Point of I[01] st X = ].
  a, b .[ holds X is open
proof
  let X be Subset of I[01], a, b be Point of I[01];
A1: 0 <= a by BORSUK_1:43;
  1 in the carrier of I[01] by BORSUK_1:43;
  then reconsider B = [. b, 1 .] as Subset of I[01] by BORSUK_1:40
,XXREAL_2:def 12;
  0 in the carrier of I[01] by BORSUK_1:43;
  then reconsider A = [. 0, a .] as Subset of I[01] by BORSUK_1:40
,XXREAL_2:def 12;
A2: b <= 1 by BORSUK_1:43;
A3: B is closed by Th20;
A4: A is closed by Th20;
  assume X = ]. a, b .[;
  then X = (A \/ B)` by A1,A2,BORSUK_1:40,XXREAL_1:200;
  hence thesis by A4,A3;
end;
