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 Th80:
  TS is deterministic implies (==>.-relation(TS) reduces x, [y1, z
  ] & ==>.-relation(TS) reduces x, [y2, z] implies y1 = y2)
proof
  assume
A1: TS is deterministic;
  assume that
A2: ==>.-relation(TS) reduces x, [y1, z] and
A3: ==>.-relation(TS) reduces x, [y2, z];
  consider P being RedSequence of ==>.-relation(TS) such that
A4: P.1 = x and
A5: P.len P = [y1, z] by A2,REWRITE1:def 3;
  consider Q being RedSequence of ==>.-relation(TS) such that
A6: Q.1 = x and
A7: Q.len Q = [y2, z] by A3,REWRITE1:def 3;
A8: (Q.len Q)`2 = z by A7;
  (P.len P)`2 = z by A5;
  then P = Q by A1,A4,A6,A8,Th69;
  hence thesis by A5,A7,XTUPLE_0:1;
end;
