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 Th69:
  for g being Element of H.i st g <> 1_(H.i)
  holds factorization [* i,g *] = <* [* i,g *] *>
proof
  let g be Element of H.i;
  assume A1: g <> 1_(H.i);
  A2: [i,g] in dom (uncurry injection H) &
    (uncurry injection H).[i,g] = [* i,g *]
  proof
    i in I;
    then A3: i in dom injection H by PARTFUN1:def 2;
    A4: injection(H,i) = (injection H).i by Def9;
    A5: dom injection(H,i) = the carrier of H.i by FUNCT_2:def 1;
    A6: [i,g] in FreeAtoms(H) by Th9;
    uncurry injection H is ManySortedSet of FreeAtoms(H) by Lm5;
    hence [i,g] in dom uncurry injection H by A6, PARTFUN1:def 2;
    thus (uncurry injection H).[i,g] = (uncurry injection H).(i,g)
      by BINOP_1:def 1
      .= injection(H,i).g by A3, A4, A5, FUNCT_5:38
      .= [* i,g *] by Th56;
  end;
  thus factorization [* i,g *] = (uncurry injection H)*<* [i,g] *> by A1, Th64
    .= <* [* i,g *] *> by A2, FINSEQ_2:34;
end;
