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 Th27:
  for g,h being Element of G.i
  holds [<*[i,g],[i,h]*>, <*[i,g*h]*>] in ReductionRel(G)
proof
  let g,h be Element of G.i;
  set p = <*>FreeAtoms(G);
  [p^<*[i,g],[i,h]*>^p, p^<*[i,g*h]*>^p]
     = [<*[i,g],[i,h]*>^p, p^<*[i,g*h]*>^p] by FINSEQ_1:34
    .= [<*[i,g],[i,h]*>, p^<*[i,g*h]*>^p] by FINSEQ_1:34
    .= [<*[i,g],[i,h]*>, <*[i,g*h]*>^p] by FINSEQ_1:34
    .= [<*[i,g],[i,h]*>, <*[i,g*h]*>] by FINSEQ_1:34;
  hence thesis by Th26;
end;
