reserve x,y,z for object, X for set, I for non empty set, i,j for Element of I,
    M0 for multMagma-yielding Function,
    M for non empty multMagma-yielding Function,
    M1, M2, M3 for non empty multMagma,
    G for Group-like multMagma-Family of I,
    H for Group-like associative multMagma-Family of I;
reserve p, q for FinSequence of FreeAtoms(H), g,h for Element of H.i,
  k for Nat;
reserve s,t for Element of FreeProduct(H);

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;
