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

theorem Th53:
  IAA = { p where p is Point of [:I[01], I[01]:] : p`2 <= 1 - 2 * (p`1) }
proof
  set P = { p where p is Point of [:I[01], I[01]:] : p`2 <= 1 - 2 * (p`1) };
  thus IAA c= P
  proof
    let x be object;
    assume
A1: x in IAA;
    then reconsider x9 = x as Point of [:I[01], I[01]:];
    consider a, b being Point of I[01] such that
A2: x = [a,b] and
A3: b <= 1 - 2 * a by A1,Def8;
    x9`1 = a & x9`2 = b by A2;
    hence thesis by A3;
  end;
  let x be object;
  assume x in P;
  then consider p being Point of [:I[01], I[01]:] such that
A4: p = x and
A5: p`2 <= 1 - 2 * (p`1);
  x in the carrier of [:I[01], I[01]:] by A4;
  then x in [:the carrier of I[01], the carrier of I[01]:] by BORSUK_1:def 2;
  then
A6: x = [x`1, x`2] by MCART_1:21;
  p`1 is Point of I[01] & p`2 is Point of I[01] by Th27;
  hence thesis by A4,A5,A6,Def8;
end;
