reserve T for non empty TopSpace;
reserve A for Subset of T;

theorem Th59:
  card (Kurat14ClPart KurExSet) = 6
proof
  set A = KurExSet;
A1: Cl (Cl (Cl A`)`)` = ]. -infty, 4 .] by Th20,BORSUK_5:51;
  5 in {5} by TARSKI:def 1;
  then 5 in Cl A` by Th17,XBOOLE_0:def 3;
  then
A2: Cl Int A = Cl (Cl A`)` & Cl A` <> Cl (Cl (Cl A`)`)` by A1,TOPS_1:def 1
,XXREAL_1:234;
  ( not 5 in Cl (Cl (Cl A`)`)`)& 5 in [. 2,+infty .[ by Th21,XXREAL_1:234,236;
  then
A3: Cl A <> Cl (Cl (Cl A`)`)` by Th10,XBOOLE_0:def 3;
  ( not 5 in Cl (Cl (Cl A`)`)`)& Cl (Cl A`)` = [. 4,+infty .[ by Th18,Th21,
BORSUK_5:55,XXREAL_1:234;
  then
A4: Cl (Cl A`)` <> Cl (Cl (Cl A`)`)` by XXREAL_1:236;
  5 in Cl (Cl (Cl A)`)` by Th14,XXREAL_1:236;
  then
A5: Cl (Cl (Cl A)`)` <> Cl (Cl (Cl A`)`)` by Th21,XXREAL_1:234;
  ( not 6 in ]. -infty, 4 .])& not 6 in {5} by TARSKI:def 1,XXREAL_1:234;
  then
A6: not 6 in Cl A` by Th17,XBOOLE_0:def 3;
  Cl (Cl (Cl A)`)` = [. 2,+infty .[ by Th13,BORSUK_5:49;
  then
A7: Cl (Cl (Cl A)`)` <> Cl A` by A6,XXREAL_1:236;
A8: 4 in Cl (Cl (Cl A)`)` & Cl Int Cl A = Cl (Cl (Cl A)`)` by Th14,TOPS_1:def 1
,XXREAL_1:236;
A9: not 4 in Cl (Cl A)` by Th12,XXREAL_1:234;
  then Cl (Cl A)` <> Cl (Cl (Cl A`)`)` by A1,XXREAL_1:234;
  then
  Cl A, Cl (Cl A)`, Cl (Cl (Cl A)`)`, Cl A`, Cl (Cl A`)`, Cl (Cl (Cl A`)`
  )` are_mutually_distinct by A9,A3,A7,A5,A8,A2,A4,Th29,Th31;
  hence thesis by BORSUK_5:3;
end;
