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 Th30:
  dom ReductionRel(G) c= FreeAtoms(G)* & rng ReductionRel(G) = FreeAtoms(G)* &
    field ReductionRel(G) = FreeAtoms(G)*
proof
  dom ReductionRel(G) c= the carrier of FreeAtoms(G)*+^+<0>;
  hence A1: dom ReductionRel(G) c= FreeAtoms(G)* by MONOID_0:61;
  rng ReductionRel(G) c= the carrier of FreeAtoms(G)*+^+<0>;
  then A2: rng ReductionRel(G) c= FreeAtoms(G)* by MONOID_0:61;
  now
    let x be object;
    assume x in FreeAtoms(G)*;
    then reconsider p = x as FinSequence of FreeAtoms(G) by FINSEQ_1:def 11;
    set q = <*>FreeAtoms(G), i = the Element of I;
    [p^<*[i,1_(G.i)]*>^q, p^q] in ReductionRel(G) by Th28;
    then p^q in rng ReductionRel(G) by XTUPLE_0:def 13;
    hence x in rng ReductionRel(G) by FINSEQ_1:34;
  end;
  then FreeAtoms(G)* c= rng ReductionRel(G) by TARSKI:def 3;
  hence A3: rng ReductionRel(G) = FreeAtoms(G)* by A2, XBOOLE_0:def 10;
  dom ReductionRel(G) \/ rng ReductionRel(G) = FreeAtoms(G)*
    by A1, A3, XBOOLE_1:12;
  hence thesis by RELAT_1:def 6;
end;
