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

theorem Th5:
  for X being Subset of I[01], a being Point of I[01] st X = [. 0, a .[
  holds X is open
proof
  let X be Subset of I[01], a be Point of I[01] such that
A1: X = [. 0, a .[;
  set Y = [. a,1 .];
  Y c= the carrier of I[01]
  proof
    let x be object;
A2: 0 <= a by BORSUK_1:43;
    assume
A3: x in Y;
    then reconsider x as Real;
    x <= 1 & a <= x by A3,XXREAL_1:1;
    hence thesis by A2,BORSUK_1:43;
  end;
  then reconsider Y as Subset of I[01];
  Y is closed & X` = Y by A1,Th3,BORSUK_4:23;
  hence thesis by TOPS_1:4;
end;
