
theorem
  for G being connected _Graph
  st ex v1,v2 being Vertex of G, e being object, H being addEdge of G,v1,e,v2
    st not e in the_Edges_of G & for W1, W2 being Walk of H
      st W1 is Cycle-like & W2 is Cycle-like holds W1.edges() = W2.edges()
  holds G is Tree-like
proof
  let G be connected _Graph;
  given v1,v2 being Vertex of G, e being object, H being addEdge of G,v1,e,v2
    such that A1: not e in the_Edges_of G and
      A2: for W1, W2 being Walk of H st W1 is Cycle-like & W2 is Cycle-like
        holds W1.edges() = W2.edges();
  G is acyclic
  proof
    assume G is non acyclic;
    then consider W1 being Walk of G such that
      A3: W1 is Cycle-like by GLIB_002:def 2;
    reconsider W3 = W1 as Walk of H by GLIB_006:75;
    A4: W3 is Cycle-like by A3, GLIB_006:24;
    e DJoins v1,v2,H by A1, GLIB_006:105;
    then A5: e Joins v2,v1,H by GLIB_000:16;
    not e in W1.edges() by A1;
    then A6: not e in W3.edges() by GLIB_001:110;
    per cases;
    suppose A7: v1 <> v2;
      consider W2 being Walk of G such that
        A8: W2 is_Walk_from v1,v2 by GLIB_002:def 1;
      set P2 = the Path-like Subwalk of W2;
      reconsider P4 = P2 as Walk of H by GLIB_006:75;
      set W4 = P4.addEdge(e);
      P2 is_Walk_from v1,v2 by A8, GLIB_001:160;
      then P4 is_Walk_from v1,v2 by GLIB_001:19;
      then A9: P4 is Path-like & P4 is_Walk_from v1,v2 by GLIB_006:23;
      then A10: W4 is closed by A5, GLIB_001:66, GLIB_001:119;
      A11: P4.first() = v1 & P4.last() = v2 by A9, GLIB_001:def 23;
      then A12: W4 is non trivial by A5, GLIB_001:132;
      A13: P4 is open by A7, A11, GLIB_001:def 24;
      not e in P2.edges() by A1;
      then A14: not e in P4.edges() by GLIB_001:110;
      for n being odd Element of NAT st 1 < n & n <= len P4 holds P4.n <> v1
       by POLYFORM:4, GLIB_001:def 28, A7, A11;
      then A16: W4 is Path-like by A5, A9, A11, A13, A14, GLIB_001:150;
      e in {e} by TARSKI:def 1;
      then e in P4.edges() \/ {e} by XBOOLE_0:def 3;
      then e in W4.edges() by A5, A11, GLIB_001:111;
      hence contradiction by A2, A4, A6, A10, A12, A16;
    end;
    suppose A17: v1 = v2;
      set W4 = H.walkOf(v2,e,v1);
      W4.edges() = {e} by A5, GLIB_001:108;
      then A18: e in W4.edges() by TARSKI:def 1;
      W4 is Cycle-like by A5, A17, GLIB_001:156;
      hence contradiction by A2, A4, A6, A18;
    end;
  end;
  hence thesis;
end;
