If any majority protest is against only a portion of the improvement or acquisition then all further proceedings to construct that portion of the improvement or to acquire that portion of the acquisition so protested against shall be barred for a period of one year, but the legislative body shall not be barred from commencing new proceedings not including any part of the improvement or acquisition so protested against. Nothing in this act contained shall prohibit the legislative body, within said one-year period, from commencing and carrying on new proceedings for the construction of a portion of the improvement or the acquiring of a portion of the acquisition so protested against if it finds, by the affirmative vote of four-fifths of its members, that the owners of a majority of the property within the area of the assessment district to be established under said new proceedings are in favor of going forward with said portion of the improvement or acquisition.