reserve G, G2 for _Graph, V, E for set,
  v for object;

theorem Th69:
  for G2, v, V for G1 being addAdjVertexAll of G2, v, V, W being Walk of G1
  for v1, v2 being Vertex of G2
  st V c= the_Vertices_of G2 & not v in the_Vertices_of G2 &
    W.first() = v1 & W.last() = v2 & not v2 in G2.reachableFrom(v1)
  holds v in W.vertices()
proof
  let G2,v,V;
  let G1 be addAdjVertexAll of G2,v,V;
  let W be Walk of G1, v1,v2 be Vertex of G2;
  assume that
    A1: V c= the_Vertices_of G2 & not v in the_Vertices_of G2 and
    A2: W.first() = v1 & W.last() = v2 & not v2 in G2.reachableFrom(v1);
  assume not v in W.vertices();
  then reconsider W2 = W as Walk of G2 by A1, Th64;
  W2.first() = v1 & W2.last() = v2 by A2, GLIB_001:16;
  then W2 is_Walk_from v1,v2 by GLIB_001:def 23;
  hence contradiction by A2, GLIB_002:def 5;
end;
