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 Th68:
  TS is deterministic implies for P, Q being RedSequence of
  ==>.-relation(TS) st P.1 = Q.1 & len P = len Q holds P = Q
proof
  assume
A1: TS is deterministic;
  let P, Q be RedSequence of ==>.-relation(TS) such that
A2: P.1 = Q.1 and
A3: len P = len Q;
  now
    let k;
    assume
A4: k in dom P;
    then 1 <= k & k <= len P by FINSEQ_3:25;
    then k in dom Q by A3,FINSEQ_3:25;
    hence P.k = Q.k by A1,A2,A4,Th67;
  end;
  hence thesis by A3,FINSEQ_2:9;
end;
