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
  PP_and(PP_imp(p,r),PP_imp(q,r)) = PP_imp(PP_or(p,q),r)
  proof
    thus PP_and(PP_imp(p,r),PP_imp(q,r))
     = PP_not PP_or(PP_and(p,PP_not(r)),PP_and(q,PP_not(r)))
    .= PP_not PP_and(PP_not(r),PP_or(p,q)) by Th30
    .= PP_imp(PP_or(p,q),r);
  end;
