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 Th61:
  t = Class(EqCl ReductionRel H,(nf s)/^k) implies nf t = (nf s)/^k
proof
  assume A1: t = Class(EqCl ReductionRel H,(nf s)/^k);
  (nf s)/^k in FreeAtoms(H)* by FINSEQ_1:def 11;
  then (nf s)/^k is Element of FreeAtoms(H)*+^+<0> by MONOID_0:61;
  then A2: (nf s)/^k in t by A1, EQREL_1:23;
  (nf s)/^k is_a_normal_form_wrt ReductionRel H
  proof
    assume not (nf s)/^k is_a_normal_form_wrt ReductionRel H;
    then consider b being object such that
      A3: [(nf s)/^k, b] in ReductionRel H by REWRITE1:def 5;
    b in rng ReductionRel(H) by A3, XTUPLE_0:def 13;
    then b in the carrier of FreeAtoms(H)*+^+<0>;
    then b in FreeAtoms(H)* by MONOID_0:61;
    then reconsider b as FinSequence of FreeAtoms(H) by FINSEQ_1:def 11;
    per cases by A3, Def3;
    suppose ex r,u being FinSequence of FreeAtoms(H), i being Element of I
        st (nf s)/^k = r^<* [i,1_(H.i)] *>^u & b = r^u;
      then consider r,u being FinSequence of FreeAtoms(H), i being Element of I
        such that A4: (nf s)/^k = r^<* [i,1_(H.i)] *>^u & b = r^u;
      A5: nf s = ((nf s)|k)^(r^<* [i,1_(H.i)] *>^u) by A4, RFINSEQ:8
        .= ((nf s)|k)^(r^<* [i,1_(H.i)] *>)^u by FINSEQ_1:32
        .= ((nf s)|k)^r^<* [i,1_(H.i)] *>^u by FINSEQ_1:32;
      set b9 = ((nf s)|k)^r^u;
      reconsider b9 as FinSequence of FreeAtoms(H);
      [nf s, b9] in ReductionRel H by A5, Def3;
      hence contradiction by Def7, REWRITE1:def 5;
    end;
    suppose ex r,u being FinSequence of FreeAtoms(H), i being Element of I,
          g,h being Element of H.i
        st (nf s)/^k = r^<* [i,g],[i,h] *>^u & b = r^<* [i,g*h] *>^u;
      then consider r,u being FinSequence of FreeAtoms(H),
          i being Element of I, g,h being Element of H.i such that
        A6: (nf s)/^k = r^<* [i,g],[i,h] *>^u & b = r^<* [i,g*h] *>^u;
      A7: nf s = ((nf s)|k)^(r^<* [i,g],[i,h] *>^u) by A6, RFINSEQ:8
        .= ((nf s)|k)^(r^<* [i,g],[i,h] *>)^u by FINSEQ_1:32
        .= ((nf s)|k)^r^<* [i,g],[i,h] *>^u by FINSEQ_1:32;
      set b9 = ((nf s)|k)^r^<* [i,g*h] *>^u;
      r^<* [i,g*h] *> is FinSequence of FreeAtoms(H) by A6, FINSEQ_1:36;
      then <* [i,g*h] *> is FinSequence of FreeAtoms(H) by FINSEQ_1:36;
      then reconsider b9 as FinSequence of FreeAtoms(H) by FINSEQ_1:75;
      [nf s, b9] in ReductionRel H by A7, Def3;
      hence contradiction by Def7, REWRITE1:def 5;
    end;
  end;
  hence thesis by A2, Def7;
end;
