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 Th36:
  for p,q being FinSequence of FreeAtoms(G) st ReductionRel(G) reduces p,q
  holds p = q or len q < len p
proof
  let p,q be FinSequence of FreeAtoms(G);
  assume A1: ReductionRel(G) reduces p,q & p <> q;
  then consider r being RedSequence of ReductionRel(G) such that
    A2: r.1 = p & r.len r = q by REWRITE1:def 3;
  A3: len r <> 1 by A1, A2;
  now
    let x be object;
    assume A4: x in dom r;
    then reconsider k = x as Nat;
    1 <= k & k <= len r by A4, FINSEQ_3:25;
    then per cases by XXREAL_0:1;
    suppose k < len r;
      then A5: k+1 <= len r by NAT_1:13;
      1+0 <= k + 1 by XREAL_1:7;
      then k+1 in dom r by A5, FINSEQ_3:25;
      then [r.k,r.(k+1)] in ReductionRel(G) by A4, REWRITE1:def 2;
      hence r.x is FinSequence by Th31;
    end;
    suppose A6: k = len r;
      then reconsider k9 = k-1 as Nat by INT_1:74;
      A7: 2-1 <= k9 by A3, A6, NAT_1:23, XREAL_1:9;
      k9 <= len r - 0 by A6, XREAL_1:10;
      then k9 in dom r by A7, FINSEQ_3:25;
      then [r.k9,r.(k9+1)] in ReductionRel(G) by A4, REWRITE1:def 2;
      hence r.x is FinSequence by Th31;
    end;
  end;
  then reconsider r as FinSequence-yielding FinSequence by PRE_POLY:def 3;
  defpred P[Nat] means $1 < len r implies len(r.($1+1)) + $1 = len p;
  A8: P[0] by A2;
  A9: for k being Nat st P[k] holds P[k+1]
  proof
    let k be Nat;
    assume A10: P[k] & k+1 < len r;
    then k+1-1 < len r - 0 by XREAL_1:14;
    then A11: len p = len(r.(k+1)) + k by A10;
    A12: k+1+1 <= len r by A10, NAT_1:13;
    A13: k+1+0 <= k+1+1 by XREAL_1:6;
    A14: 0+1 <= k+1 by XREAL_1:6;
    then 1 <= k+1+1 & k+1 <= len r by A12, A13, XXREAL_0:2;
    then k+1 in dom r & k+1+1 in dom r by A12, A14, FINSEQ_3:25;
    then [r.(k+1),r.(k+1+1)] in ReductionRel(G) by REWRITE1:def 2;
    then len(r.(k+1)) = len(r.(k+1+1)) + 1 by Th35;
    hence thesis by A11;
  end;
  A15: for k being Nat holds P[k] from NAT_1:sch 2(A8,A9);
  reconsider k = len r - 1 as Nat by INT_1:74;
  k < len r - 0 by XREAL_1:15;
  then A16: len p = len(r.(k+1)) + k by A15
    .= len q + k by A2;
  assume len q >= len p;
  then len q + 0 >= len q + k by A16;
  then k = 0 by XREAL_1:6;
  hence contradiction by A3;
end;
