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 Th37:
  {} is_a_normal_form_wrt ReductionRel G
proof
  now
    set p = <*>FreeAtoms(G);
    given q being object such that
      A1: [p,q] in ReductionRel(G);
    q in field ReductionRel(G) by A1, RELAT_1:15;
    then q in FreeAtoms(G)* by Th30;
    then reconsider q as FinSequence of FreeAtoms(G) by FINSEQ_1:def 11;
    len p = len q + 1 by A1, Th35;
    hence contradiction;
  end;
  hence thesis by REWRITE1:def 5;
end;
