
theorem Th4:
  for A,B being non empty set, F,G being Function of B,A for I
being Indexing of F,G for m being Element of B holds I`2.m is Functor of I`1.(F
  .m), I`1.(G.m)
proof
  let A,B be non empty set, F,G be Function of B,A;
  let I be Indexing of F,G;
  reconsider H = I`2 as ManySortedFunctor of I`1*F, I`1*G by Def8;
  let m be Element of B;
  dom (I`1*F) = B by PARTFUN1:def 2;
  then
A1: (I`1*F).m = I`1.(F.m) by FUNCT_1:12;
  H.m is Functor of (I`1*F).m, (I`1*G).m & dom (I`1*G) = B by PARTFUN1:def 2;
  hence thesis by A1,FUNCT_1:12;
end;
