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 Th42:
  I is trivial implies for p being non empty FinSequence of FreeAtoms(G)
    ex g being Element of G.i st ReductionRel(G) reduces p,<*[i,g]*>
proof
  assume A1: I is trivial;
  defpred P[Nat] means for p being non empty FinSequence of FreeAtoms(G)
    st len p = $1 + 1 ex g being Element of G.i
    st ReductionRel(G) reduces p,<*[i,g]*>;
  A2: P[0]
  proof
    let p be non empty FinSequence of FreeAtoms(G);
    assume A3: len p = 0 + 1;
    then A4: p = <* p.1 *> by FINSEQ_1:40;
    1 in dom p by A3, FINSEQ_3:25;
    then A5: p.1 in FreeAtoms(G) by FINSEQ_2:11;
    then consider x,y being object such that
      A6: p.1 = [x,y] by RELAT_1:def 1;
    x in dom G by A5, A6, Th7;
    then A7: x = i by A1, ZFMISC_1:def 10;
    then reconsider g = y as Element of G.i by A5, A6, Th9;
    take g;
    thus thesis by A4, A6, A7, REWRITE1:12;
  end;
  A8: for k being Nat st P[k] holds P[k+1]
  proof
    let k be Nat;
    assume A9: P[k];
    let p be non empty FinSequence of FreeAtoms(G);
    assume A10: len p = (k+1)+1;
    len p <> 0;
    then consider q being FinSequence of FreeAtoms(G),
        d being Element of FreeAtoms(G) such that
      A11: p = q^<*d*> by FINSEQ_2:19;
    len p = len q + 1 by A11, FINSEQ_2:16;
    then A12: len q = k+1 by A10;
    then q is non empty;
    then consider h being Element of G.i such that
      A13: ReductionRel(G) reduces q,<*[i,h]*> by A9, A12;
    consider x,y being object such that
      A14: d = [x,y] by RELAT_1:def 1;
    x in dom G by A14, Th7;
    then A15: x = i by A1, ZFMISC_1:def 10;
    then reconsider g = y as Element of G.i by A14, Th9;
    A16: <*[i,g]*> is FinSequence of FreeAtoms(G) &
      <*[i,h]*> is FinSequence of FreeAtoms(G) by Th23;
    ReductionRel(G) reduces <*d*>,<*[i,g]*> by A14, A15, REWRITE1:12;
    then ReductionRel(G) reduces q^<*d*>,<*[i,h]*>^<*[i,g]*> by A13, A16, Th41;
    then A17: ReductionRel(G) reduces p,<*[i,h],[i,g]*> by A11, FINSEQ_1:def 9;
    take h*g;
    [<*[i,h],[i,g]*>, <*[i,h*g]*>] in ReductionRel(G) by Th27;
    then ReductionRel(G) reduces <*[i,h],[i,g]*>, <*[i,h*g]*> by REWRITE1:15;
    hence thesis by A17, REWRITE1:16;
  end;
  A18: for k being Nat holds P[k] from NAT_1:sch 2(A2,A8);
  let p be non empty FinSequence of FreeAtoms(G);
  0+1 < len p + 1 by XREAL_1:8;
  then 1 <= len p by NAT_1:13;
  then consider k being Nat such that
    A19: len p = 1 + k by NAT_1:10;
  thus thesis by A18, A19;
end;
