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

theorem Th3:
  for X being Subset of I[01], a being Point of I[01] st X = [. 0, a .[
  holds X` = [. a, 1 .]
proof
  set I = the carrier of I[01];
  let X be Subset of I[01], a be Point of I[01] such that
A1: X = [. 0, a .[;
  set Y = [. a,1 .];
A2: X` = I \ X by SUBSET_1:def 4;
  hereby
    let x be object;
    assume
A3: x in X`;
    then reconsider y = x as Point of I[01];
    not x in X by A2,A3,XBOOLE_0:def 5;
    then
A4: y >= a or y < 0 by A1,XXREAL_1:3;
    y <= 1 by BORSUK_1:43;
    hence x in Y by A4,BORSUK_1:43,XXREAL_1:1;
  end;
  let x be object;
  assume
A5: x in Y;
  then reconsider y = x as Real;
A6: a <= y by A5,XXREAL_1:1;
  then
A7: not y in X by A1,XXREAL_1:3;
A8: 0 <= a by BORSUK_1:43;
  y <= 1 by A5,XXREAL_1:1;
  then y in I by A6,A8,BORSUK_1:43;
  hence thesis by A2,A7,XBOOLE_0:def 5;
end;
