theorem
  E is epsilon-transitive implies (E |= the_axiom_of_pairs iff for X,Y
  st X in E & Y in E holds { X,Y } in E)
proof
  assume
A1: E is epsilon-transitive;
  hence E |= the_axiom_of_pairs implies for X,Y st X in E & Y in E holds { X,Y
  } in E by Th2;
  assume for X,Y st X in E & Y in E holds { X,Y } in E;
  then for u,v holds { u,v } in E;
  hence thesis by A1,Th2;
end;
