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 Th34:
  (len p = len q & for k, i, g st p.k = [i,g] holds (Rev q).k = [i,g"])
  implies ReductionRel(H) reduces p^q,{} & ReductionRel(H) reduces q^p,{}
proof
  assume that A1: len p = len q and
    A2: for k, i, g st p.k = [i,g] holds (Rev q).k = [i,g"];
  for k, i, g, h st p.k = [i,g] & g*h = 1_(H.i) holds
    (Rev q).k = [i,h] by A2, Lm1;
  hence ReductionRel(H) reduces p^q,{} by A1, Th33;
  for k, i, h st q.k = [i,h] holds (Rev p).k = [i,h"]
  proof
    let k, i, h;
    assume A3: q.k = [i,h];
    then A4: k in dom q by FUNCT_1:def 2;
    then k in Seg len p by A1, FINSEQ_1:def 3;
    then A5: len p - k + 1 in Seg len p by FINSEQ_5:2;
    then reconsider m = len p - k + 1 as Nat;
    A6: m in dom p by A5, FINSEQ_1:def 3;
    then p.m in rng p by FUNCT_1:3;
    then A7: p.m in FreeAtoms(H);
    then consider i9,g being object such that
      A8: p.m = [i9,g] by RELAT_1:def 1;
    i9 in dom H by A7, A8, Th7;
    then reconsider i9 as Element of I;
    reconsider g as Element of H.i9 by A7, A8, Th9;
    A9: m in dom q by A1, A6, FINSEQ_3:29;
    [i9,g"] = (Rev q).m by A2, A8
      .= q.(len q - m + 1) by A9, FINSEQ_5:58
      .= [i,h] by A1, A3;
    then A10: i9 = i & g" = h by XTUPLE_0:1;
    then reconsider g as Element of H.i;
    dom p = dom q by A1, FINSEQ_3:29;
    hence (Rev p).k = [i,h"] by A4, A8, A10, FINSEQ_5:58;
  end;
  then for k, i, h, g st q.k = [i,h] & h*g = 1_(H.i) holds
    (Rev p).k = [i,g] by Lm1;
  hence thesis by A1, Th33;
end;
