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 Th6:
  d9 in d & d in Collapse (E,A) implies d9 in Collapse (E,A) & ex B
  st B in A & d9 in Collapse (E,B)
proof
  assume that
A1: d9 in d and
A2: d in Collapse (E,A);
  d in { d1 : for d st d in d1 ex B st B in A & d in Collapse (E,B) } by A2,Th1
;
  then
  ex d1 st d = d1 & for d st d in d1 ex B st B in A & d in Collapse (E,B);
  then consider B such that
A3: B in A and
A4: d9 in Collapse (E,B) by A1;
  Collapse (E,B) c= Collapse (E,A) by Th4,A3,ORDINAL1:def 2;
  hence d9 in Collapse (E,A) by A4;
  thus thesis by A3,A4;
end;
