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