theorem Th54:
  for g being Element of G.i holds <*[i,g]*> =
    (commute <*<:(the carrier of G.i)-->i,id the carrier of(G.i):>*>).g
proof
  let g be Element of G.i;
  set C = the carrier of G.i;
  A1: dom <:C-->i,id C :> = C by Lm3;
  thus (commute <*<:C-->i,id C :>*>).g
     = <*<:C-->i,id C :>.g*> by A1, Th4
    .= <*[(C-->i).g,(id C).g]*> by A1, FUNCT_3:def 7
    .= <*[i,g]*>;
end;
