reserve p, q, x, y for Real,
  n for Nat;

theorem Th4:
  for X being Subset of I[01], a being Point of I[01] st X = ]. a, 1 .]
  holds X is open
proof
  let X be Subset of I[01], a be Point of I[01] such that
A1: X = ]. a, 1 .];
  set Y = [. 0,a .];
  Y c= the carrier of I[01]
  proof
    let x be object;
A2: a <= 1 by BORSUK_1:43;
    assume
A3: x in Y;
    then reconsider x as Real;
A4: 0 <= x by A3,XXREAL_1:1;
    x <= a by A3,XXREAL_1:1;
    then x <= 1 by A2,XXREAL_0:2;
    hence thesis by A4,BORSUK_1:43;
  end;
  then reconsider Y as Subset of I[01];
  Y is closed & X` = Y by A1,Th2,BORSUK_4:23;
  hence thesis by TOPS_1:4;
end;
