reserve x, x1, x2, y, y1, y2, z, z1, z2 for object, X, X1, X2 for set;
reserve E for non empty set;
reserve e for Element of E;
reserve u, u9, u1, u2, v, v1, v2, w, w1, w2 for Element of E^omega;
reserve F, F1, F2 for Subset of E^omega;
reserve i, k, l, n for Nat;
reserve TS for non empty transition-system over F;
reserve s, s9, s1, s2, t, t1, t2 for Element of TS;
reserve S for Subset of TS;

theorem Th66:
  not <%>E in rng dom (the Tran of TS) implies for P being
RedSequence of ==>.-relation(TS), k, l st k in dom P & l in dom P & k < l holds
  (P.k)`2 <> (P.l)`2
proof
  defpred P[Nat] means for P being RedSequence of ==>.-relation(TS), k, l st
  len P = $1 & k in dom P & l in dom P & k < l holds (P.k)`2 <> (P.l)`2;
  assume
A1: not <%>E in rng dom (the Tran of TS);
A2: now
    let i;
    assume
A3: P[i];
    now
      let P be RedSequence of ==>.-relation(TS), k, l such that
A4:   len P = i + 1 and
A5:   k in dom P and
A6:   l in dom P and
A7:   k < l;
A8:   i < len P by A4,NAT_1:13;
A9:   k <= len P by A5,FINSEQ_3:25;
A10:  1 <= k by A5,FINSEQ_3:25;
A11:  1 <= l by A6,FINSEQ_3:25;
A12:  l <= len P by A6,FINSEQ_3:25;
      per cases;
      suppose
        k = 1 & l = len P;
        hence (P.k)`2 <> (P.l)`2 by A1,A7,Th61;
      end;
      suppose
A13:    k <> 1 & l = len P;
        reconsider k1 = k - 1 as Nat by A10,NAT_1:21;
A14:    k > 1 by A10,A13,XXREAL_0:1;
        then k1 > 1 - 1 by XREAL_1:9;
        then
A15:    k1 >= 0 + 1 by NAT_1:13;
        reconsider l1 = l - 1 as Nat by A11,NAT_1:21;
A16:    k1 < l1 by A7,XREAL_1:9;
A17:    l > 1 by A7,A10,A11,XXREAL_0:1;
        then consider Q being RedSequence of ==>.-relation(TS) such that
A18:    <*P.1*>^Q = P and
A19:    len Q + 1 = len P by A13,Th5;
        l1 > 1 - 1 by A17,XREAL_1:9;
        then
A20:    l1 >= 0 + 1 by NAT_1:13;
        k1 <= len Q + 1 - 1 by A9,A19,XREAL_1:9;
        then
A21:    k1 in dom Q by A15,FINSEQ_3:25;
A22:    len <*P.1*> = 1 by FINSEQ_1:40;
        then
A23:    P.l = (Q.l1) by A12,A17,A18,FINSEQ_1:24;
        l1 <= len Q + 1 - 1 by A12,A19,XREAL_1:9;
        then
A24:    l1 in dom Q by A20,FINSEQ_3:25;
        P.k = (Q.k1) by A9,A14,A18,A22,FINSEQ_1:24;
        hence (P.k)`2 <> (P.l)`2 by A3,A4,A19,A21,A24,A16,A23;
      end;
      suppose
A25:    l <> len P;
        k < i + 1 by A4,A7,A12,XXREAL_0:2;
        then
A26:    k <= i by NAT_1:13;
        then reconsider Q = P | i as RedSequence of ==>.-relation(TS) by A10,
REWRITE2:3,XXREAL_0:2;
A27:    P.k = Q.k by A26,FINSEQ_3:112;
        l < i + 1 by A4,A12,A25,XXREAL_0:1;
        then
A28:    l <= i by NAT_1:13;
        then
A29:    P.l = Q.l by FINSEQ_3:112;
        k <= len Q by A8,A26,FINSEQ_1:59;
        then
A30:    k in dom Q by A10,FINSEQ_3:25;
        l <= len Q by A8,A28,FINSEQ_1:59;
        then
A31:    l in dom Q by A11,FINSEQ_3:25;
        len Q = i by A8,FINSEQ_1:59;
        hence (P.k)`2 <> (P.l)`2 by A3,A7,A30,A31,A27,A29;
      end;
    end;
    hence P[i + 1];
  end;
A32: P[0];
A33: for i holds P[i] from NAT_1:sch 2(A32, A2);
  let P be RedSequence of ==>.-relation(TS), k, l such that
A34: k in dom P & l in dom P & k < l;
  len P = len P;
  hence thesis by A33,A34;
end;
