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

theorem Th56:
  IAA \/ IBB \/ ICC = [:[.0,1.], [.0,1.]:]
proof
  thus IAA \/ IBB \/ ICC c= [:[.0,1.], [.0,1.]:] by Th1;
  let x be object;
  assume
A1: x in [:[.0,1.], [.0,1.]:];
  then reconsider q = x`1, p = x`2 as Point of I[01] by BORSUK_1:40,MCART_1:10;
A2: x = [q,p] by A1,MCART_1:21;
  x in IAA or x in IBB or x in ICC
  proof
    per cases;
    suppose
A3:   p >= 1 - 2 * q;
      now
        per cases;
        suppose
          p >= 2 * q - 1;
          hence thesis by A2,A3,Def9;
        end;
        suppose
          p < 2 * q - 1;
          hence thesis by A2,Def10;
        end;
      end;
      hence thesis;
    end;
    suppose
      p < 1 - 2 * q;
      hence thesis by A2,Def8;
    end;
  end;
  then x in IAA \/ IBB or x in ICC by XBOOLE_0:def 3;
  hence thesis by XBOOLE_0:def 3;
end;
