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
  d /\ E c= Collapse (E,A) iff d in Collapse (E,succ A)
proof
A1: Collapse (E,succ A) = { d9 : for d1 st d1 in d9 ex B st B in succ A & d1
  in Collapse (E,B) } by Th1;
  thus d /\ E c= Collapse (E,A) implies d in Collapse (E,succ A)
  proof
    assume
A2: for a being object st a in d /\ E holds a in Collapse (E,A);
    now
      let d1;
      assume d1 in d;
      then d1 in d /\ E by XBOOLE_0:def 4;
      then d1 in Collapse (E,A) by A2;
      hence ex B st B in succ A & d1 in Collapse (E,B) by ORDINAL1:6;
    end;
    hence thesis by A1;
  end;
  assume d in Collapse (E,succ A);
  then
A3: ex e1 being Element of E st e1 = d & for d1 st d1 in e1 ex B st B in
  succ A & d1 in Collapse (E,B) by A1;
  let a be object;
  assume
A4: a in d /\ E;
  then reconsider e = a as Element of E by XBOOLE_0:def 4;
  a in d by A4,XBOOLE_0:def 4;
  then consider B such that
A5: B in succ A and
A6: e in Collapse (E,B) by A3;
A7: now
    Collapse (E,B) = { d9 : for d1 st d1 in d9 ex C st C in B & d1 in
    Collapse (E,C) } by Th1;
    then
A8: ex d9 st d9 = e & for d1 st d1 in d9 ex C st C in B & d1 in Collapse
    (E,C) by A6;
    let d1;
    assume d1 in e;
    then consider C such that
A9: C in B & d1 in Collapse (E,C) by A8;
    take C;
    B c= A by A5,ORDINAL1:22;
    hence C in A & d1 in Collapse (E,C) by A9;
  end;
  Collapse (E,A) = { d9 : for d1 st d1 in d9 ex B st B in A & d1 in
  Collapse (E,B) } by Th1;
  hence thesis by A7;
end;
