reserve k,n,m for Nat,
  A,B,C for Ordinal,
  X for set,
  x,y,z for object;
reserve f,g,h,fx for Function,
  K,M,N for Cardinal,
  phi,psi for
  Ordinal-Sequence;
reserve a,b for Aleph;

theorem Th22:
  cf nextcard a = nextcard a
proof
  nextcard a is_cofinal_with cf nextcard a by Def1;
  then consider xi being Ordinal-Sequence such that
A1: dom xi = cf nextcard a and
A2: rng xi c= nextcard a and
  xi is increasing and
A3: nextcard a = sup xi;
A4: card Union xi c= Sum Card xi & Sum (cf nextcard a --> a) = (cf nextcard
  a)*` a by CARD_2:65,CARD_3:39;
A5: card nextcard a = nextcard a & succ Union xi = (Union xi) +^ 1 by
ORDINAL2:31;
A6: now
    let x be object;
    assume
A7: x in cf nextcard a;
    xi.x in rng xi by A1,A7,FUNCT_1:def 3;
    then
A8: card (xi.x) in nextcard a by A2,CARD_1:8,ORDINAL1:12;
    (Card xi).x = card (xi.x) & (cf nextcard a --> a).x = a by A1,A7,
CARD_3:def 2,FUNCOP_1:7;
    hence (Card xi).x c= (cf nextcard a --> a).x by A8,CARD_3:91;
  end;
  dom Card xi = dom xi & dom (cf nextcard a --> a) = cf nextcard a by
CARD_3:def 2;
  then Sum Card xi c= Sum (cf nextcard a --> a) by A1,A6,CARD_3:30;
  then card Union xi c= (cf nextcard a)*`a by A4;
  then
A9: (card Union xi) +` 1 c= (cf nextcard a)*`a +` 1 by CARD_2:84;
A10: card ((Union xi) +^ 1) = (card Union xi) +` card 1 by CARD_2:13;
  ex A st rng xi c= A by ORDINAL2:def 4;
  then On rng xi = rng xi by ORDINAL3:6;
  then
A11: card nextcard a c= card succ Union xi by A3,CARD_1:11,ORDINAL3:72;
A12: cf nextcard a c= nextcard a by Def1;
  now
    per cases;
    suppose
      cf nextcard a = 0;
      then
A13:  (cf nextcard a)*`a = 0 by CARD_2:20;
      thus thesis by A11,A5,A9,A10,A13;
    end;
    suppose
A14:  cf nextcard a <> 0;
      0 c= cf nextcard a;
      then
A15:  0 in cf nextcard a by A14,CARD_1:3;
A16:  cf nextcard a c= a or a c= cf nextcard a;
      1 in a by Lm1,Th15;
      then
A17:  (cf nextcard a)*`a = a & a+`1 = a & a in nextcard a or (cf nextcard
a)*`a = cf nextcard a & cf nextcard a is Aleph by A15,A16,Th17,CARD_1:18
,CARD_2:76,CARD_4:16;
      then
      nextcard a c= (cf nextcard a) +` 1 & 1 in cf nextcard a by A11,A5,A9,A10
,Th15,CARD_1:4;
      then nextcard a c= cf nextcard a by A11,A5,A9,A10,A17,CARD_1:4
,CARD_2:76;
      hence thesis by A12;
    end;
  end;
  hence thesis;
end;
