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 Th31:
  for x,y being object st [x,y] in ReductionRel(G)
  holds x is FinSequence of FreeAtoms(G) & y is FinSequence of FreeAtoms(G)
proof
  let x,y be object;
  assume [x,y] in ReductionRel(G);
  then x in field ReductionRel(G) & y in field ReductionRel(G) by RELAT_1:15;
  then x in FreeAtoms(G)* & y in FreeAtoms(G)* by Th30;
  hence thesis by FINSEQ_1:def 11;
end;
