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 Th40:
  for g being Element of H.i, h being Element of H.j
  holds <*[i,g]*>,<*[j,h]*> are_convertible_wrt ReductionRel H iff
    (g = 1_(H.i) & h = 1_(H.j)) or (i = j & g = h)
proof
  let g be Element of H.i, h being Element of H.j;
  [i,g] in FreeAtoms(H) & [j,h] in FreeAtoms(H) by Th9;
  then reconsider s1=<*[i,g]*>, s2=<*[j,h]*> as FinSequence of FreeAtoms(H)
    by FINSEQ_1:74;
  hereby
    assume <*[i,g]*>,<*[j,h]*> are_convertible_wrt ReductionRel H;
    then s1,s2 are_convergent_wrt ReductionRel H by REWRITE1:def 23;
    then consider c being object such that
      A1: ReductionRel H reduces <*[i,g]*>,c and
      A2: ReductionRel H reduces <*[j,h]*>,c by REWRITE1:def 7;
    per cases;
    suppose A3: c = s1;
      len s1 = 1 & len s2 = 1 by FINSEQ_1:40;
      then [i,g] = [j,h] by A2, A3, Th36, FINSEQ_1:76;
      hence (g = 1_(H.i) & h = 1_(H.j)) or (i = j & g = h) by XTUPLE_0:1;
    end;
    suppose A4: c <> s1;
      then c in field ReductionRel H by A1, REWRITE1:18;
      then c in FreeAtoms(H)* by Th30;
      then reconsider c as FinSequence of FreeAtoms(H) by FINSEQ_1:def 11;
      len c < len s1 by A1, A4, Th36;
      then len c < 1 by FINSEQ_1:40;
      then c <> s2 by FINSEQ_1:40;
      hence (g = 1_(H.i) & h = 1_(H.j)) or (i = j & g = h)
        by A1, A2, A4, Th38, REWRITE1:33;
    end;
  end;
  assume (g = 1_(H.i) & h = 1_(H.j)) or (i = j & g = h);
  then per cases;
  suppose g = 1_(H.i) & h = 1_(H.j);
    then [s1,{}] in ReductionRel(H) & [s2,{}] in ReductionRel(H) by Th29;
    then A5: s1,{} are_convertible_wrt ReductionRel(H) &
      s2,{} are_convertible_wrt ReductionRel(H) by REWRITE1:29;
    then {},s2 are_convertible_wrt ReductionRel(H) by REWRITE1:31;
    hence thesis by A5, REWRITE1:30;
  end;
  suppose i = j & g = h;
    hence thesis by REWRITE1:26;
  end;
end;
