let i be object ; :: according to FINSET_1:def 5 :: thesis: ( not i in I or (F .:.: A) . i is finite )
assume A1: i in I ; :: thesis: (F .:.: A) . i is finite
reconsider f = F . i as Function ;
f .: (A . i) is finite ;
hence (F .:.: A) . i is finite by A1, PBOOLE:def 20; :: thesis: verum