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 Th22:
  ex q st len p = len q &
    for k, i, g st p.k = [i,g] holds (Rev q).k = [i,g"]
proof
  consider q such that
    A1: len p = len q and
    A2: for k being Nat, i being Element of I, g being Element of H.i
      st p.k = [i,g] ex h being Element of H.i
      st g*h = 1_(H.i) & (Rev q).k = [i,h] by Th21;
  take q;
  thus len p = len q by A1;
  let k, i, g;
  assume p.k = [i,g];
  then consider h being Element of H.i such that
    A3: g*h = 1_(H.i) & (Rev q).k = [i,h] by A2;
  thus thesis by A3, GROUP_1:12;
end;
