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 Th44:
  TS is deterministic implies ([x, y1] in ==>.-relation(TS) & [x,
  y2] in ==>.-relation(TS) implies y1 = y2)
proof
  assume
A1: TS is deterministic;
  assume that
A2: [x, y1] in ==>.-relation(TS) and
A3: [x, y2] in ==>.-relation(TS);
  consider s2, v2, t2, w2 such that
  x = [s2, v2] and
A4: y2 = [t2, w2] by A3,Th31;
  consider s1, v1, t1, w1 such that
A5: x = [s1, v1] and
A6: y1 = [t1, w1] by A2,Th31;
A7: s1, v1 ==>. t1, w1, TS by A2,A5,A6,Def4;
A8: s1, v1 ==>. t2, w2, TS by A3,A5,A4,Def4;
  then t1 = t2 by A1,A7,Th30;
  hence thesis by A1,A6,A4,A7,A8,Th30;
end;
