reserve X,Y,Z for set,
  x,y,z for object,
  E for non empty set,
  A,B,C for Ordinal ,
  L,L1 for Sequence,
  f,f1,f2,h for Function,
  d,d1,d2,d9 for Element of E;

theorem Th8:
  ex A st E = Collapse (E,A)
proof
  defpred R[object,object] means
    ex A st $2 = A & $1 in Collapse(E,A) & for B st $1
  in Collapse(E,B) holds A c= B;
A1: now
    let x be object;
    assume x in E;
    then reconsider d = x as Element of E;
    defpred Q[Ordinal] means d in Collapse(E,$1);
A2: ex A st Q[A] by Th5;
    ex A st Q[A] & for B st Q[B] holds A c= B from ORDINAL1:sch 1 (A2);
    hence ex y being object st R[x,y];
  end;
  consider f such that
A3: dom f = E &
     for x being object st x in E holds R[x,f.x] from CLASSES1:sch 1(A1);
  x in rng f implies x is Ordinal
  proof
    assume x in rng f;
    then consider y being object such that
A4: y in dom f and
A5: x = f.y by FUNCT_1:def 3;
    ex A st f.y = A & y in Collapse (E,A) & for C st y in Collapse (E,C)
    holds A c= C by A3,A4;
    hence thesis by A5;
  end;
  then consider A such that
A6: rng f c= A by ORDINAL1:24;
  take A;
  thus for x being object holds x in E implies x in Collapse (E,A)
  proof let x be object;
    assume
A7: x in E;
    then consider B such that
A8: f.x = B and
A9: x in Collapse (E,B) and
    for C st x in Collapse (E,C) holds B c= C by A3;
    B in rng f by A3,A7,A8,FUNCT_1:def 3;
    then Collapse (E,B) c= Collapse (E,A) by Th4,A6,ORDINAL1:def 2;
    hence thesis by A9;
  end;
  thus thesis by Th7;
end;
