reserve T for non empty TopSpace,
  a, b, c, d for Point of T;

theorem Th71:
  for x being Point of I[01] st x < 1/2 holds not [x,0] in IBB &
  not [x,0] in ICC
proof
  let x be Point of I[01];
  assume
A1: x < 1/2;
  thus not [x,0] in IBB
  proof
    assume [x,0] in IBB;
    then consider a, b being Point of I[01] such that
A2: [x,0] = [a,b] and
A3: b >= 1 - 2 * a and
    b >= 2 * a - 1 by Def9;
    x = a & b = 0 by A2,XTUPLE_0:1;
    then 0 + 2 * x >= 1 by A3,XREAL_1:20;
    then (2 * x)/2 >= 1/2 by XREAL_1:72;
    hence thesis by A1;
  end;
  not [x,0] in ICC
  proof
    assume [x,0] in ICC;
    then consider a, b being Point of I[01] such that
A4: [x,0] = [a,b] and
A5: b <= 2 * a - 1 by Def10;
    x = a & b = 0 by A4,XTUPLE_0:1;
    then 0 + 1 <= 2 * x by A5,XREAL_1:19;
    then 1/2 <= (2 * x)/2 by XREAL_1:72;
    hence thesis by A1;
  end;
  hence thesis;
end;
