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 Th65:
  len nf s = 1 implies ex i, g st g <> 1_(H.i) & s = [* i, g *]
proof
  assume A1: len nf s = 1;
  then 1 in dom nf s by FINSEQ_3:25;
  then (nf s).1 in rng nf s by FUNCT_1:3;
  then reconsider z = (nf s).1 as Element of FreeAtoms(H);
  consider x,y being object such that
    A2: z = [x,y] by RELAT_1:def 1;
  x in dom H by A2, Th7;
  then reconsider i = x as Element of I;
  reconsider g = y as Element of H.i by A2, Th9;
  take i, g;
  A3: nf s = <* [i,g] *> by A1, A2, FINSEQ_1:40;
  thus g <> 1_(H.i)
  proof
    assume g = 1_(H.i);
    then A4: <* [i, 1_(H.i)] *> is_a_normal_form_wrt ReductionRel H
      by A3, Def7;
    [<* [i, 1_(H.i)] *>, {}] in ReductionRel H by Th29;
    hence contradiction by A4, REWRITE1:def 5;
  end;
  consider p being Element of FreeAtoms(H)*+^+<0> such that
    A5: s = Class(EqCl ReductionRel H,p) by EQREL_1:36;
  <* [i,g] *> in s by A3, Def7;
  hence thesis by A5, EQREL_1:23;
end;
