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 Th32:
  x in dom(PP_imp(p,q)) implies
   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
  proof
    assume
A1: x in dom(PP_imp(p,q));
    dom(PP_imp(p,q)) =
    {d where d is Element of D:
    d in dom p & p.d = FALSE or d in dom q & q.d = TRUE
    or d in dom p & p.d = TRUE & d in dom q & q.d = FALSE} by Th31;
    then ex d1 being Element of D st d1 = x &
    (d1 in dom p & p.d1 = FALSE or d1 in dom q & q.d1 = TRUE
    or d1 in dom p & p.d1 = TRUE & d1 in dom q & q.d1 = FALSE) by A1;
    hence thesis;
  end;
