Real property belonging to, or held in trust for, any organization mentioned in subdivision (7), (10), (11), (13), (14), (15), (16) or (18) of section 12-81, which real property is so held for one or more of the purposes stated in the applicable subdivision, and from which real property no rents, profits or income are derived, shall be exempt from taxation though not in actual use therefor by reason of the absence of suitable buildings and improvements thereon, if the construction of such buildings or improvements is in progress. The real property belonging to, or held in trust for, any such organization, not used exclusively for carrying out one or more of such purposes but leased, rented or otherwise used for other purposes, shall not be exempt. If a portion only of any lot or building belonging to, or held in trust for, any such organization is used exclusively for carrying out one or more of such purposes, such lot or building shall be so exempt only to the extent of the portion so used and the remaining portion shall be subject to taxation.