
theorem Th25:
  { p where p is Point of [:I[01],I[01]:] : p`2 >= 1 - 2 * (p`1) &
  p`2 >= 2 * (p`1) - 1 } is closed non empty Subset of [:I[01],I[01]:]
proof
  set GG = [:I[01],I[01]:], SS = [:R^1,R^1:];
  1 in the carrier of I[01] by BORSUK_1:43;
  then [1,1] in [:the carrier of I[01], the carrier of I[01]:] by ZFMISC_1:87;
  then reconsider x = [1,1] as Point of GG by BORSUK_1:def 2;
  reconsider PA = { p where p is Point of SS : p`2 >= 1 - 2 * (p`1) & p`2 >= 2
  * (p`1) - 1 } as closed Subset of SS by Th23;
  set P0 = { p where p is Point of GG : p`2 >= 1 - 2 * (p`1) & p`2 >= 2 * (p`1
  ) - 1 };
A1: x`2 >= 2 * (x`1) - 1;
A2: GG is SubSpace of SS by BORSUK_3:21;
A3: P0 = PA /\ [#] GG
  proof
    thus P0 c= PA /\ [#] GG
    proof
      let x be object;
A4:   the carrier of GG c= the carrier of SS by A2,BORSUK_1:1;
      assume x in P0;
      then
A5:   ex p being Point of GG st x = p & p`2 >= 1 - 2 * (p`1) & p`2 >= 2 *
      (p`1) - 1;
      then x in the carrier of GG;
      then reconsider a = x as Point of SS by A4;
      a`2 >= 1 - 2 * (a`1) by A5;
      then x in PA by A5;
      hence thesis by A5,XBOOLE_0:def 4;
    end;
    let x be object;
    assume
A6: x in PA /\ [#] GG;
    then x in PA by XBOOLE_0:def 4;
    then
    ex p being Point of SS st x = p & p`2 >= 1 - 2 * (p`1) & p`2 >= 2 * (
    p`1) - 1;
    hence thesis by A6;
  end;
  x`2 >= 1 - 2 * (x`1);
  then x in P0 by A1;
  hence thesis by A2,A3,PRE_TOPC:13;
end;
