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;

theorem Th38:
  for g being Element of G.i st g <> 1_(G.i)
  holds <* [i,g] *> is_a_normal_form_wrt ReductionRel G
proof
  let g be Element of G.i;
  assume A1: g <> 1_(G.i);
  now
    set p = <* [i,g] *>;
    given q being object such that
      A2: [p,q] in ReductionRel G;
    p in field ReductionRel G & q in field ReductionRel G by A2, RELAT_1:15;
    then p in FreeAtoms(G)* & q in FreeAtoms(G)* by Th30;
    then reconsider p,q as FinSequence of FreeAtoms(G) by FINSEQ_1:def 11;
    per cases by A2, Def3;
    suppose ex s,t being FinSequence of FreeAtoms(G), j being Element of I
        st p = s^<* [j,1_(G.j)] *>^t & q = s^t;
      then consider s,t being FinSequence of FreeAtoms(G), j being Element of I
        such that A3: p = s^<* [j,1_(G.j)] *>^t & q = s^t;
      0+1 = len p by FINSEQ_1:40
        .= len(s^<* [j,1_(G.j)] *>) + len t by A3, FINSEQ_1:22
        .= len s + 1 + len t by FINSEQ_2:16;
      then len s + len t = 0;
      then A4: s = {} & t = {};
      then p = <* [j,1_(G.j)] *>^t by A3, FINSEQ_1:34
        .= <* [j,1_(G.j)] *> by A4, FINSEQ_1:34;
      then [i,g] = [j,1_(G.j)] by FINSEQ_1:76;
      then i = j & g = 1_(G.j) by XTUPLE_0:1;
      hence contradiction by A1;
    end;
    suppose ex s,t being FinSequence of FreeAtoms(G), j being Element of I,
          h1,h2 being Element of (G.j)
        st p = s^<* [j,h1],[j,h2] *>^t & q = s^<* [j,h1*h2] *>^t;
      then consider s,t being FinSequence of FreeAtoms(G),
          j being Element of I, h1,h2 being Element of (G.j)
        such that A5: p = s^<* [j,h1],[j,h2] *>^t & q = s^<* [j,h1*h2] *>^t;
      0+1 = len p by FINSEQ_1:40
        .= len(s^<* [j,h1],[j,h2] *>) + len t by A5, FINSEQ_1:22
        .= len s + len <* [j,h1],[j,h2] *> + len t by FINSEQ_1:22
        .= len s + 2 + len t by FINSEQ_1:44
        .= len s + len t + 1 + 1;
      hence contradiction;
    end;
  end;
  hence thesis by REWRITE1:def 5;
end;
