theorem Th39:
  F is commutative associative & F is having_a_unity & e =
the_unity_wrt F & G is_distributive_wrt F & G.(e,d) = e implies G.(F"**"p,d) =
  F "**"(G[:](p,d))
proof
  assume that
A1: F is commutative associative & F is having_a_unity & e =
  the_unity_wrt F and
A2: G is_distributive_wrt F and
A3: G.(e,d) = e;
A4: len p = len(G[:](p,d)) & Seg len p = dom p by FINSEQ_1:def 3,FINSEQ_2:84;
A5: Seg len(G[:](p,d)) = dom(G[:](p,d)) by FINSEQ_1:def 3;
  thus G.(F"**"p,d) = G.(F$$(findom p,[#](p,e)),d) by A1,Def2
    .= F $$(findom p,G[:]([#](p,e),d)) by A1,A2,A3,Th13
    .= F $$(findom p,[#](G[:](p,d),e)) by A3,Lm5
    .= F "**"(G[:](p,d)) by A1,A4,A5,Def2;
end;
