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 Th1:
  Collapse (E,A) = { d : for d1 st d1 in d ex B st B in A & d1 in
  Collapse (E,B) }
proof
  defpred P[object,object] means ex B st B = $1 & $2 = Collapse (E,B);
  deffunc H(Sequence) = { d : for d1 st d1 in d ex C st C in dom($1) & d1 in
  union { $1.C } };
  deffunc F(Ordinal) = Collapse(E,$1);
A1: for x being object st x in A ex y being object st P[x,y]
  proof
    let x be object;
    assume x in A;
    then reconsider B = x as Ordinal;
    take Collapse (E,B) , B;
    thus thesis;
  end;
  consider f such that
A2: dom f = A & for x being object st x in A holds P[x,f.x]
         from CLASSES1:sch 1(A1);
  reconsider L = f as Sequence by A2,ORDINAL1:def 7;
A3: now
    let A;
    assume A in dom L;
    then ex B st B = A & L.A = Collapse (E,B) by A2;
    hence L.A = F(A);
  end;
A4: for A for x being object
    holds x = F(A) iff ex L st x = H(L) & dom L = A & for B st B in
  A holds L.B = H(L|B) by Def1;
  for B st B in dom L holds L.B = H(L|B) from ORDINAL1:sch 5(A4,A3);
  then
A5: Collapse (E,A) = { d : for d1 st d1 in d ex B st B in dom L & d1 in
  union { L.B } } by A2,Def1;
  now
    let x be object;
    assume
    x in { d : for d1 st d1 in d ex B st B in dom L & d1 in union { L .B } };
    then consider d such that
A6: x = d and
A7: for d1 st d1 in d ex B st B in dom L & d1 in union { L.B };
    for d1 st d1 in d ex B st B in A & d1 in Collapse (E,B)
    proof
      let d1;
      assume d1 in d;
      then consider B such that
A8:   B in dom L and
A9:   d1 in union { L.B } by A7;
      take B;
      thus B in A by A2,A8;
      L.B = Collapse (E,B) by A3,A8;
      hence thesis by A9,ZFMISC_1:25;
    end;
    hence x in { d1 : for d st d in d1 ex B st B in A & d in Collapse (E,B) }
    by A6;
  end;
  hence Collapse (E,A) c= { d : for d1 st d1 in d ex B st B in A & d1 in
  Collapse (E,B) } by A5;
  let x be object;
  assume x in { d1 : for d st d in d1 ex B st B in A & d in Collapse (E,B) };
  then consider d1 such that
A10: x = d1 and
A11: for d st d in d1 ex B st B in A & d in Collapse (E,B);
  for d st d in d1 ex B st B in dom L & d in union { L.B }
  proof
    let d;
    assume d in d1;
    then consider B such that
A12: B in A and
A13: d in Collapse (E,B) by A11;
    take B;
    thus B in dom L by A2,A12;
    L.B = Collapse (E,B) by A2,A3,A12;
    hence thesis by A13,ZFMISC_1:25;
  end;
  hence thesis by A5,A10;
end;
