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

theorem Th58:
  ICC /\ IBB = { p where p is Point of [:I[01], I[01]:] : p`2 = 2 * (p`1) - 1 }
proof
  set KK = { p where p is Point of [:I[01], I[01]:] : p`2 = 2 * (p`1) - 1 };
  thus ICC /\ IBB c= KK
  proof
    let x be object;
    assume
A1: x in ICC /\ IBB;
    then x in ICC by XBOOLE_0:def 4;
    then consider p being Point of [:I[01], I[01]:] such that
A2: x = p and
A3: p`2 <= 2 * (p`1) - 1 by Th55;
    x in IBB by A1,XBOOLE_0:def 4;
    then
    ex q being Point of [:I[01], I[01]:] st x = q & q`2 >= 1 - 2 * (q`1) &
    q`2 >= 2 * (q`1) - 1 by Th54;
    then p`2 = 2 * (p`1) - 1 by A2,A3,XXREAL_0:1;
    hence thesis by A2;
  end;
  let x be object;
  assume x in KK;
  then consider p being Point of [:I[01], I[01]:] such that
A4: p = x and
A5: p`2 = 2 * (p`1) - 1;
  x in the carrier of [:I[01], I[01]:] by A4;
  then
A6: x in [:the carrier of I[01], the carrier of I[01]:] by BORSUK_1:def 2;
  then
A7: x = [p`1,p`2] & p`1 in the carrier of I[01] by A4,MCART_1:10,21;
A8: p`2 in the carrier of I[01] by A4,A6,MCART_1:10;
  then
A9: x in ICC by A5,A7,Def10;
  2 * p`1 - 1 >= 0 by A5,A8,BORSUK_1:43;
  then 2 * p`1 >= 0 + 1 by XREAL_1:19;
  then (2 * p`1)/2 >= 1/2 by XREAL_1:72;
  then 2 * p`1 - 1 >= 0 & 0 >= 1 - 2 * (p`1) by XREAL_1:219,220;
  then x in IBB by A5,A7,A8,Def9;
  hence thesis by A9,XBOOLE_0:def 4;
end;
