Every city or authority employe otherwise eligible to join the pension fund created under the act to which this is a supplement but who is ineligible to join because of not having done so within the times provided by such act shall have the option of joining such pension fund within one year after the effective date of this act. Any city or authority employe who joins such pension plan shall receive full credit toward his pension under the act for each year of service as an employe of the city or authority upon his producing proof, satisfactory to the pension board, of the number of years of such service and upon his making back payments at the same rate as if he had been a city employe and a member of the pension fund during such period in a lump sum or by installments as may be determined by the board. If the employe is retired before the back payments have been completed, the annuity shall be reduced by an amount equivalent to the unpaid back payments or deduction increases not anticipated.