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

theorem Th64:
  for G2, v, V for G1 being addAdjVertexAll of G2,v,V, W being Walk of G1
  st V c= the_Vertices_of G2 & not v in the_Vertices_of G2 &
    ((W.edges() c= the_Edges_of G2 & W is non trivial)
      or not v in W.vertices())
  holds W is Walk of G2
proof
  let G2, v, V;
  let G1 be addAdjVertexAll of G2,v,V;
  let W be Walk of G1;
  assume that
    A1: V c= the_Vertices_of G2 & not v in the_Vertices_of G2 and
    A2: (W.edges() c= the_Edges_of G2 & W is non trivial)
      or not v in W.vertices();
  A3: W.edges() c= the_Edges_of G2 & not v in W.vertices() by A1, A2, Th63;
  for w being object holds w in W.vertices() implies w in the_Vertices_of G2
  proof
    let w be object;
    assume A4: w in W.vertices();
    then A5: w in the_Vertices_of G1;
    assume A6: not w in the_Vertices_of G2;
    the_Vertices_of G1 = the_Vertices_of G2 \/ {v} by A1, Def4;
    then w in {v} by A5, A6, XBOOLE_0:def 3;
    hence contradiction by A4, A3, TARSKI:def 1;
  end;
  then A7: W.vertices() c= the_Vertices_of G2 by TARSKI:def 3;
  rng W = W.vertices() \/ W.edges() by GLIB_001:101;
  then A8: W is FinSequence of the_Vertices_of G2 \/ the_Edges_of G2
    by FINSEQ_1:def 4,A7, A3, XBOOLE_1:13;
  now
    thus len W is odd;
    W.first() in W.vertices() by GLIB_001:88;
    then W.1 in W.vertices() by GLIB_001:def 6;
    hence W.1 in the_Vertices_of G2 by A7;
    let n be odd Element of NAT;
    assume A9: n < len W;
    then A10: W.(n+1) Joins W.n,W.(n+2),G1 by GLIB_001:def 3;
    W.(n+1) in W.edges() by A9, GLIB_001:100;
    hence W.(n+1) Joins W.n,W.(n+2),G2 by A10, GLIB_006:72, A3;
  end;
  hence thesis by A8, GLIB_001:def 3;
end;
