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 Th56:
  injection(H,i).g = [* i, g *]
proof
  set C = the carrier of H.i;
  A1: <*[i,g]*> in [* i,g *] by Th48;
  dom commute <*<:C-->i,id C :>*> = C by Th53;
  hence injection(H,i).g = (proj Class EqCl ReductionRel H).
      ((commute <*<:C-->i,id C :>*>).g) by FUNCT_1:13
    .= (proj Class EqCl ReductionRel H).<*[i,g]*> by Th54
    .= [* i,g *] by A1, EQREL_1:65;
end;
