theorem Th12:
  F is commutative associative & F is having_a_unity & e =
the_unity_wrt F & G is_distributive_wrt F & G.(d,e) = e implies G.(d,F$$(B,f))
  = F $$(B,G[;](d,f))
proof
  assume that
A1: F is commutative associative & F is having_a_unity and
A2: e = the_unity_wrt F and
A3: G is_distributive_wrt F;
  defpred X[Element of Fin C] means G.(d,F$$($1,f)) = F $$($1,G[;](d,f));
A4: for B9 being Element of Fin C, b being Element of C holds X[B9] & not b
  in B9 implies X[B9 \/ {.b.}]
  proof
    let B9,c such that
A5: G.(d,F$$(B9,f)) = F $$(B9,G[;](d,f)) and
A6: not c in B9;
    thus G.(d,F$$(B9 \/ {.c.},f)) = G.(d,F.(F$$(B9,f),f.c)) by A1,A6,Th2
      .= F.(G.(d,F$$(B9,f)),G.(d,f.c)) by A3,BINOP_1:11
      .= F.(F $$(B9,G[;](d,f)),(G[;](d,f)).c) by A5,FUNCOP_1:53
      .= F $$(B9 \/ {.c.},G[;](d,f)) by A1,A6,Th2;
  end;
  assume G.(d,e) = e;
  then G.(d,F$$({}.C,f)) = e by A1,A2,SETWISEO:31
    .= F $$({}.C,G[;](d,f)) by A1,A2,SETWISEO:31;
  then
A7: X[{}.C];
  for B holds X[B] from SETWISEO:sch 2(A7,A4);
  hence thesis;
end;
