reserve m for Cardinal,
  A,B,C for Ordinal,
  x,y,z,X,Y,Z,W for set,
  f for Function;
reserve f,g for Function,
  L for Sequence,
  F for Cardinal-Function;

theorem Th36:
  W is Tarski & X c= Rank card W implies X,Rank card W
  are_equipotent or X in Rank card W
proof
  assume that
A1: W is Tarski and
A2: X c= Rank card W and
A3: not X,Rank card W are_equipotent;
  defpred P[object] means ex Y st Y in X & $1 = the_rank_of Y;
  consider LL being set such that
A4: for x being object holds x in LL iff x in On W & P[x] from XBOOLE_0:sch 1;
  consider ff being Cardinal-Function such that
A5: dom ff = LL & for x st x in LL holds ff.x = g(x) from CARD_3:sch 1;
A6: LL c= On W
  by A4;
A7: product ff c= Funcs(LL,W)
  proof
    let x be object;
    assume x in product ff;
    then consider g such that
A8: x = g and
A9: dom g = dom ff and
A10: for x being object st x in dom ff holds g.x in ff.x by CARD_3:def 5;
    rng g c= W
    proof
      let y be object;
      assume y in rng g;
      then consider x being object such that
A11:  x in dom g and
A12:  y = g.x by FUNCT_1:def 3;
      reconsider x as set by TARSKI:1;
A13:  ff.x = card bool x by A5,A9,A11;
      x in W by A6,A5,A9,A11,ORDINAL1:def 9;
      then bool x in W by A1;
      then card bool x in W by A1,Th11;
      then
A14:  card bool x c= W by A1,Th5;
      y in ff.x by A9,A10,A11,A12;
      hence thesis by A13,A14;
    end;
    hence thesis by A5,A8,A9,FUNCT_2:def 2;
  end;
A15: card W = card Rank card W by A1,Th34;
  then
A16: card X <> card W by A3,CARD_1:5;
  On W c= W by ORDINAL2:7;
  then
A17: LL c= W by A6;
  now
    let Z;
    assume Z in union LL;
    then consider Y such that
A18: Z in Y and
A19: Y in LL by TARSKI:def 4;
    Y in On W by A4,A19;
    then reconsider Y as Ordinal by ORDINAL1:def 9;
A20: Y c= union LL by A19,ZFMISC_1:74;
A21: Z in Y by A18;
    hence Z is Ordinal;
    reconsider A = Z as Ordinal by A21;
    A c= Y by A18,ORDINAL1:def 2;
    hence Z c= union LL by A20;
  end;
  then reconsider ULL = union LL as epsilon-transitive epsilon-connected set
by ORDINAL1:19;
A22: dom Card id LL = dom id LL by CARD_3:def 2;
A23: dom id LL = LL by RELAT_1:45;
  now
    let x be object;
    assume
A24: x in dom Card id LL;
    then
A25: (Card id LL).x = card ((id LL).x) by A22,CARD_3:def 2;
A26: (id LL).x = x by A23,A22,A24,FUNCT_1:18;
    reconsider xx=x as set by TARSKI:1;
    ff.x = card bool xx by A5,A23,A22,A24;
    hence (Card id LL).x in ff.x by A26,A25,CARD_1:14;
  end;
  then
A27: Sum Card id LL in Product ff by A5,A23,A22,CARD_3:41;
  consider f such that
A28: dom f = X &
for x being object st x in X holds f.x = f(x) from FUNCT_1:sch 3;
  LL c= rng f
  proof
    let x be object;
    assume x in LL;
    then consider Y such that
A29: Y in X and
A30: x = the_rank_of Y by A4;
    f.Y = x by A28,A29,A30;
    hence thesis by A28,A29,FUNCT_1:def 3;
  end;
  then
A31: card LL c= card X by A28,CARD_1:12;
A32: card product ff = Product ff by CARD_3:def 8;
  card X c= card W by A2,A15,CARD_1:11;
  then card X in card W by A16,CARD_1:3;
  then card LL <> card W by A31,ORDINAL1:12;
  then not LL,W are_equipotent by CARD_1:5;
  then LL in W by A1,A17;
  then Funcs(LL,W) c= W by A1,A17,Th22;
  then product ff c= W by A7;
  then
A33: Product ff c= card W by A32,CARD_1:11;
A34: card W is limit_ordinal by A1,Th19;
A35: card W = On W by A1,Th9;
  X c= Rank succ ULL
  proof
    let x be object;
            reconsider xx=x as set by TARSKI:1;
    defpred P[Ordinal] means $1 in card W & xx c= Rank $1;
    assume
A36: x in X;
    then
A37: f.x = the_rank_of xx by A28;
    consider A such that
A38: A in card W and
A39: x in Rank A by A2,A34,A36,CLASSES1:29,31;
    P[A] by A38,A39,ORDINAL1:def 2;
    then
A40: ex A st P[A];
    consider A such that
A41: P[A] and
A42: for B st P[B] holds A c= B from ORDINAL1:sch 1(A40);
    now
      let B;
      assume xx c= Rank B;
      then A c= card W & card W c= B or A c= B by A41,A42,ORDINAL1:16;
      hence A c= B;
    end;
    then A = the_rank_of xx by A41,CLASSES1:def 9;
    then f.x in LL by A4,A35,A36,A37,A41;
    then the_rank_of xx c= ULL by A37,ZFMISC_1:74;
    then
A43: Rank the_rank_of xx c= Rank ULL by CLASSES1:37;
    xx c= Rank the_rank_of xx by CLASSES1:def 9;
    then xx c= Rank ULL by A43;
    hence thesis by CLASSES1:32;
  end;
  then
A44: X in Rank succ succ ULL by CLASSES1:32;
  Union id LL = union rng id LL by CARD_3:def 4
    .= ULL by RELAT_1:45;
  then card ULL in card W by A27,A33,CARD_3:39,ORDINAL1:12;
  then
A45: ULL <> On W by A35;
  union card W = card W by A34;
  then ULL c= On W by A6,A35,ZFMISC_1:77;
  then ULL c< On W by A45;
  then ULL in card W by A35,ORDINAL1:11;
  then succ ULL in card W by A34,ORDINAL1:28;
  then succ succ ULL in card W by A34,ORDINAL1:28;
  hence thesis by A34,A44,CLASSES1:31;
end;
