
theorem Th24:
  { p where p is Point of [:I[01],I[01]:] : p`2 <= 1 - 2 * (p`1) }
  is closed non empty Subset of [:I[01],I[01]:]
proof
  set GG = [:I[01],I[01]:], SS = [:R^1,R^1:];
  0 in the carrier of I[01] by BORSUK_1:43;
  then [0,0] in [:the carrier of I[01], the carrier of I[01]:] by ZFMISC_1:87;
  then reconsider x = [0,0] as Point of GG by BORSUK_1:def 2;
  reconsider PA = { p where p is Point of SS : p`2 <= 1 - 2 * (p`1) } as
  closed Subset of SS by Th21;
  set P0 = { p where p is Point of GG : p`2 <= 1 - 2 * (p`1) };
A1: GG is SubSpace of SS by BORSUK_3:21;
A2: P0 = PA /\ [#] GG
  proof
    thus P0 c= PA /\ [#] GG
    proof
      let x be object;
A3:   the carrier of GG c= the carrier of SS by A1,BORSUK_1:1;
      assume x in P0;
      then
A4:   ex p being Point of GG st x = p & p`2 <= 1 - 2 * (p`1);
      then x in the carrier of GG;
      then reconsider a = x as Point of SS by A3;
      a`2 <= 1 - 2 * (a`1) by A4;
      then x in PA;
      hence thesis by A4,XBOOLE_0:def 4;
    end;
    let x be object;
    assume
A5: 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);
    hence thesis by A5;
  end;
  x`2 <= 1 - 2 * (x`1);
  then x in P0;
  hence thesis by A1,A2,PRE_TOPC:13;
end;
