reserve x,y,z for Variable,
  H for ZF-formula,
  E for non empty set,
  a,b,c,X,Y,Z for set,
  u,v,w for Element of E,
  f,g,h,i,j for Function of VAR,E;

theorem
  E is epsilon-transitive implies (E |= the_axiom_of_unions iff for X st
  X in E holds union X in E)
proof
  assume
A1: E is epsilon-transitive;
  hence E |= the_axiom_of_unions implies for X st X in E holds union X in E by
Th4;
  assume for X st X in E holds union X in E;
  then for u holds union u in E;
  hence thesis by A1,Th4;
end;
