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

theorem Th59:
  for x being Point of [:I[01], I[01]:] st x in IAA holds x`1 <= 1 /2
proof
  set GG = [:I[01], I[01]:];
  let x be Point of GG;
  assume x in IAA;
  then consider a, b being Point of I[01] such that
A1: x = [a,b] and
A2: b <= 1 - 2 * a by Def8;
  b >= 0 by BORSUK_1:43;
  then 1 >= 0 + 2 * a by A2,XREAL_1:19;
  then
A3: 1/2 >= (2 * a)/2 by XREAL_1:72;
  thus thesis by A1,A3;
end;
