reserve x, y, X for set;
reserve E for non empty set;
reserve e for Element of E;
reserve u, u1, v, v1, v2, w, w9, w1, w2 for Element of E^omega;
reserve F for Subset of E^omega;
reserve i, k, l for Nat;
reserve TS for non empty transition-system over F;
reserve S, T for Subset of TS;

theorem Th11:
  for TS being transition-system over Lex E
   holds the Tran of TS is Function implies TS is deterministic
proof
  let TS be transition-system over Lex(E) such that
A1: (the Tran of TS) is Function;
A2: now
    let p be Element of TS, u, v such that
A3: u <> v and
A4: [p, u] in dom (the Tran of TS) & [p, v] in dom (the Tran of TS);
    u in Lex(E) & v in Lex(E) by A4,ZFMISC_1:87;
    hence not ex w st u^w = v or v^w = u by A3,Th10;
  end;
  not <%>E in rng dom (the Tran of TS) by Th8;
  hence thesis by A1,A2,REWRITE3:def 1;
end;
