theorem
  F is commutative associative & F is having_a_unity & F is
having_an_inverseOp & G is_distributive_wrt F implies G[;](d,id D).(F$$(B,f)) =
  F $$(B,G[;](d,id D)*f)
proof
  assume that
A1: F is commutative associative & F is having_a_unity and
A2: F is having_an_inverseOp and
A3: G is_distributive_wrt F;
  set e = the_unity_wrt F;
  set u = G[;](d,id D);
  u is_distributive_wrt F by A3,FINSEQOP:54;
  then
A4: for d1,d2 holds u.(F.(d1,d2)) = F.(u.d1,u.d2);
  G[;](d,id D).e = e by A1,A2,A3,FINSEQOP:69;
  hence thesis by A1,A4,Th16;
end;
