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 Th13:
  len w = 1 implies (X = w-succ_of (S, TS) iff S, w ==>* X, _bool TS)
proof
  assume
A1: len w = 1;
  thus X = w-succ_of (S, TS) implies S, w ==>* X, _bool TS
  proof
    assume X = w-succ_of (S, TS);
    then [[S, w], X] in the Tran of _bool TS by A1,Def1;
    then S, w -->. X, _bool TS by REWRITE3:def 2;
    then S, w ==>. X, <%>E, _bool TS by REWRITE3:23;
    then S, w ==>* X, <%>E, _bool TS by REWRITE3:87;
    hence thesis by REWRITE3:def 7;
  end;
  assume S, w ==>* X, _bool TS;
  then
A2: S, w ==>* X, <%>E, _bool TS by REWRITE3:def 7;
  ex e st w = <%e%> & w.0 = e by A1,Th4;
  then S, w^{} ==>. X, <%>E, _bool TS by A2,Th12;
  then
A3: S, w -->. X, _bool TS by REWRITE3:24;
  then X in _bool TS by REWRITE3:15;
  then X in the carrier of _bool TS;
  then reconsider X9 = X as Subset of TS by Def1;
  [[S, w], X9] in the Tran of _bool TS by A3,REWRITE3:def 2;
  hence thesis by Def1;
end;
