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 Th50:
  for g being Element of H.i, h being Element of H.j
  holds [* i,g *] = [* j,h *] iff (g = 1_(H.i) & h = 1_(H.j)) or(i = j & g = h)
proof
  let g be Element of H.i, h be Element of H.j;
  [i,g] in FreeAtoms(H) by Th9;
  then <*[i,g]*> is FinSequence of FreeAtoms(H) by FINSEQ_1:74;
  then <*[i,g]*> in FreeAtoms(H)* by FINSEQ_1:def 11;
  then A1: <*[i,g]*> in the carrier of FreeAtoms(H)*+^+<0> by MONOID_0:61;
  [j,h] in FreeAtoms(H) by Th9;
  then <*[j,h]*> is FinSequence of FreeAtoms(H) by FINSEQ_1:74;
  then <*[j,h]*> in FreeAtoms(H)* by FINSEQ_1:def 11;
  then A2: <*[j,h]*> in the carrier of FreeAtoms(H)*+^+<0> by MONOID_0:61;
  hereby
    assume [* i,g *] = [* j,h *];
    then [<*[i,g]*>,<*[j,h]*>] in EqCl ReductionRel H by A1, EQREL_1:35;
    then <*[i,g]*>,<*[j,h]*> are_convertible_wrt ReductionRel H
      by A1, A2, MSUALG_6:41;
    hence (g = 1_(H.i) & h = 1_(H.j)) or (i = j & g = h) by Th40;
  end;
  assume (g = 1_(H.i) & h = 1_(H.j)) or (i = j & g = h);
  then <*[i,g]*>,<*[j,h]*> are_convertible_wrt ReductionRel H by Th40;
  then [<*[i,g]*>,<*[j,h]*>] in EqCl ReductionRel H by A1, A2, MSUALG_6:41;
  hence thesis by A1, EQREL_1:35;
end;
