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 Th63:
  not <%>E in rng dom (the Tran of TS) implies for P being
  RedSequence of ==>.-relation(TS) st P.1 = [x, <%e%>] & P.len P = [y, <%>E]
  holds len P = 2
proof
  assume
A1: not <%>E in rng dom (the Tran of TS);
  let P be RedSequence of ==>.-relation(TS) such that
A2: P.1 = [x, <%e%>] & P.len P = [y, <%>E];
  len P <= len <%e%> + 1 by A1,A2,Th62;
  then
 len P <= 1 + 1 by AFINSQ_1:34;
  then
A3: len P = 0 or ... or len P = 2;
  len P <> 1 by A2,XTUPLE_0:1;
  hence thesis by A3;
end;
