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;
