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 Th25:
  for p,q,r being FinSequence of FreeAtoms(G) st [p,q] in ReductionRel(G)
  holds [p^r,q^r] in ReductionRel(G) & [r^p,r^q] in ReductionRel(G)
proof
  let p,q,r be FinSequence of FreeAtoms(G);
  assume [p,q] in ReductionRel(G);
  then per cases by Def3;
  suppose ex s,t being FinSequence of FreeAtoms(G), i being Element of I
      st p = s^<* [i,1_(G.i)] *>^t & q = s^t;
    then consider s,t being FinSequence of FreeAtoms(G), i being Element of I
      such that A1: p = s^<* [i,1_(G.i)] *>^t & q = s^t;
    A2: p^r = s^<* [i,1_(G.i)] *>^(t^r) by A1, FINSEQ_1:32;
    q^r = s^(t^r) by A1, FINSEQ_1:32;
    hence [p^r,q^r] in ReductionRel(G) by A2, Def3;
    A3: r^p = (r^(s^<* [i,1_(G.i)] *>))^t by A1, FINSEQ_1:32
      .= r^s^<* [i,1_(G.i)] *>^t by FINSEQ_1:32;
    r^q = r^s^t by A1, FINSEQ_1:32;
    hence thesis by A3, Def3;
  end;
  suppose ex s,t being FinSequence of FreeAtoms(G), i being Element of I,
        g,h being Element of (G.i)
      st p = s^<* [i,g],[i,h] *>^t & q = s^<* [i,g*h] *>^t;
    then consider s,t being FinSequence of FreeAtoms(G), i being Element of I,
        g,h being Element of (G.i) such that
      A4: p = s^<* [i,g],[i,h] *>^t & q = s^<* [i,g*h] *>^t;
    A5: p^r = s^<* [i,g],[i,h] *>^(t^r) by A4, FINSEQ_1:32;
    q^r = s^<* [i,g*h] *>^(t^r) by A4, FINSEQ_1:32;
    hence [p^r,q^r] in ReductionRel(G) by A5, Def3;
    A6: r^p = (r^(s^<* [i,g],[i,h] *>))^t by A4, FINSEQ_1:32
      .= r^s^<* [i,g],[i,h] *>^t by FINSEQ_1:32;
    r^q = r^(s^<* [i,g*h] *>)^t by A4, FINSEQ_1:32
      .= r^s^<* [i,g*h] *>^t by FINSEQ_1:32;
    hence thesis by A6, Def3;
  end;
end;
