reserve x for object;
reserve D for set;
reserve p for PartialPredicate of D;
reserve D for non empty set;
reserve p,q,r for PartialPredicate of D;

theorem
  x in dom PP_imp(p,q) & PP_imp(p,q).x = FALSE implies
   x in dom p & p.x = TRUE & x in dom q & q.x = FALSE
  proof
    assume x in dom PP_imp(p,q);
    then x in dom p & p.x = FALSE or x in dom q & q.x = TRUE
    or x in dom p & p.x = TRUE & x in dom q & q.x = FALSE by Th32;
    hence thesis by Th33,Def4;
  end;
