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

theorem Th72:
  IAA /\ ICC = { [1/2,0] }
proof
  thus IAA /\ ICC c= { [1/2,0] }
  proof
    let x be object;
    assume
A1: x in IAA /\ ICC;
    then reconsider y = x as Point of [:I[01], I[01]:];
    x in IAA by A1,XBOOLE_0:def 4;
    then
A2: y`1 <= 1/2 by Th59;
A3: x in ICC by A1,XBOOLE_0:def 4;
    then y`1 >= 1/2 by Th60;
    then
A4: y`1 = 1/2 by A2,XXREAL_0:1;
    y in the carrier of [:I[01], I[01]:];
    then
A5: y in [:the carrier of I[01], the carrier of I[01]:] by BORSUK_1:def 2;
A6: y`2 is Point of I[01] by Th27;
    ex q being Point of [:I[01], I[01]:] st q = y & q`2 <= 2 * (q`1) - 1 by A3
,Th55;
    then y`2 = 0 by A4,A6,BORSUK_1:43;
    then y = [1/2,0] by A5,A4,MCART_1:21;
    hence thesis by TARSKI:def 1;
  end;
  1/2 is Point of I[01] by BORSUK_1:43;
  then [1/2,0] in IAA & [1/2,0] in ICC by Th69,Th70;
  then [1/2,0] in IAA /\ ICC by XBOOLE_0:def 4;
  hence thesis by ZFMISC_1:31;
end;
