In connection with any project financed in whole or in part by a government, the authority shall also have power to mortgage all or any part of its property, real or personal, then owned or thereafter acquired, and thereby:

(1) To vest in a government the right, upon the happening of an event of default, as defined in such mortgage, to foreclose such mortgage through judicial proceedings or through the exercise of a power of sale without judicial proceedings, so long as a government shall be the holder of any of the bonds secured by such mortgage.

(2) To vest in a trustee or trustees the right, upon the happening of an event of default, as defined in such mortgage, to foreclose such mortgage through judicial proceedings or through the exercise of a power of sale without judicial proceedings, but only with the consent of the government which aided in financing the housing project involved.

(3) To vest in other obligees the right to foreclose such mortgage by judicial proceedings, but only with the consent of the government which aided in financing the project involved.

(4) To vest in an obligee, including a government, the right in foreclosing any mortgage as provided in this section to foreclose such mortgage as to all or such part or parts of the property covered thereby as such obligee, in its absolute discretion, shall elect. The institution, prosecution, and conclusion of any such foreclosure proceedings or the sale of any such parts of the mortgaged property shall not affect in any manner or to any extent the lien of the mortgage on the parts of the mortgaged property not included in such proceedings or not sold as provided in this section.