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 Th61:
  not <%>E in rng dom (the Tran of TS) implies for P being
  RedSequence of ==>.-relation(TS) st (P.1)`2 = (P.len P)`2 holds len P = 1
proof
  assume
A1: not <%>E in rng dom (the Tran of TS);
  let P be RedSequence of ==>.-relation(TS) such that
A2: (P.1)`2 = (P.len P)`2;
  per cases;
  suppose
A3: len P <= 1;
    len P >= 0 + 1 by NAT_1:13;
    hence len P = 1 by A3,XXREAL_0:1;
  end;
  suppose
A4: len P > 1;
    then reconsider p1 = len P - 1 as Nat by NAT_1:21;
A5: p1 + 1 = len P;
    then
A6: p1 <= len P by NAT_1:13;
    1 + 1 <= len P by A4,NAT_1:13;
    then
A7: 1 + 1 in dom P by FINSEQ_3:25;
    0 + 1 <= p1 + 1 by XREAL_1:6;
    then
A8: p1 + 1 in dom P by FINSEQ_3:25;
    1 <= p1 by A4,A5,NAT_1:13;
    then p1 in dom P by A6,FINSEQ_3:25;
    then consider s2, v2, t2, w2 such that
    P.p1 = [s2, v2] and
A9: P.(p1 + 1) = [t2, w2] by A8,Th47;
    1 in dom P by A4,FINSEQ_3:25;
    then consider s1, v1, t1, w1 such that
A10: P.1 = [s1, v1] and
    P.(1 + 1) = [t1, w1] by A7,Th47;
    (P.len P)`2 = w2 by A9;
    then v1 = w2 by A2,A10;
    hence len P = 1 by A1,A10,A9,Th60;
  end;
end;
