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 Th35:
  for p,q being FinSequence st [p,q] in ReductionRel(G) holds len p = len q + 1
proof
  let p9,q9 be FinSequence;
  assume A1: [p9,q9] in ReductionRel(G);
  then reconsider p = p9, q = q9 as FinSequence of FreeAtoms(G) by Th31;
  per cases by A1, 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 A2: p = s^<* [i,1_(G.i)] *>^t & q = s^t;
    thus len p9 = len(s^<* [i,1_(G.i)] *>) + len t by A2, FINSEQ_1:22
      .= len s + 1 + len t by FINSEQ_2:16
      .= len s + len t + 1
      .= len q9 + 1 by A2, FINSEQ_1:22;
  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
      A3: p = s^<* [i,g],[i,h] *>^t & q = s^<* [i,g*h] *>^t;
    thus len p9 = len(s^<* [i,g],[i,h] *>) + len t by A3, FINSEQ_1:22
      .= len s + len<* [i,g],[i,h] *> + len t by FINSEQ_1:22
      .= len s + (1+1) + len t by FINSEQ_1:44
      .= len s + 1 + len t + 1
      .= len(s^<* [i,g*h] *>) + len t + 1 by FINSEQ_2:16
      .= len q9 + 1 by A3, FINSEQ_1:22;
  end;
end;
