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

theorem Th63:
  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
  holds
    (W.edges() c= the_Edges_of G2 & W is non trivial implies
      not v in W.vertices())
    & (not v in W.vertices() implies W.edges() c= the_Edges_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;
  consider E being set such that
    A2: card V = card E & E misses the_Edges_of G2 &
    the_Edges_of G1 = the_Edges_of G2 \/ E and
    A3: for v1 being object st v1 in V ex e1 being object st e1 in E &
      e1 Joins v1,v,G1 &
      for e2 being object st e2 Joins v1,v,G1 holds e1 = e2 by A1, Def4;
  A4: E /\ the_Edges_of G2 = {} by A2, XBOOLE_0:def 7;
  hereby
    assume A5: W.edges() c= the_Edges_of G2 & W is non trivial;
    thus not v in W.vertices()
    proof
      assume v in W.vertices();
      then consider n being odd Element of NAT such that
        A6: n <= len W & W.n = v by GLIB_001:87;
      per cases by A6, XXREAL_0:1;
      suppose A7: n = len W;
        A8: 1 <> len W by A5, GLIB_001:126;
        1 <= len W by ABIAN:12;
        then 1 < len W by A8, XXREAL_0:1;
        then reconsider m = len W - 2 as odd Element of NAT by Lm13;
        A9: m < len W - 0 by XREAL_1:15;
        then W.(m+1) Joins W.m,W.(m+2),G1 by GLIB_001:def 3;
        then A10: W.(m+1) Joins W.m,v,G1 by A6, A7;
        then W.m in V by A1, Def4;
        then consider e1 being object such that
          A11: e1 in E & e1 Joins W.m,v,G1 and
          A12: for e2 being object st e2 Joins W.m,v,G1 holds e1 = e2 by A3;
        W.(m+1) in E by A10, A11, A12;
        then not W.(m+1) in W.edges() by A5,A4,Lm7;
        hence contradiction by A9, GLIB_001:100;
      end;
      suppose A13: n < len W;
        then A14: W.(n+1) Joins W.(n+2),v,G1 by A6,GLIB_000:14,GLIB_001:def 3;
        then W.(n+2) in V by A1, Def4;
        then consider e1 being object such that
          A15: e1 in E & e1 Joins W.(n+2),v,G1 and
          A16: for e2 being object st e2 Joins W.(n+2),v,G1 holds e1 = e2 by A3
;
        W.(n+1) in E by A14, A15, A16;
        then not W.(n+1) in W.edges() by A5,A4,Lm7;
        hence contradiction by A13, GLIB_001:100;
      end;
    end;
  end;
  assume A17: not v in W.vertices();
  for e being object holds e in W.edges() implies e in the_Edges_of G2
  proof
    let e be object;
    assume e in W.edges();
    then consider n being odd Element of NAT such that
      A18: n < len W & W.(n+1) = e by GLIB_001:100;
    A19: e Joins W.n,W.(n+2),G1 by A18, GLIB_001:def 3;
    W.n <> v & W.(n+2) <> v
    proof
      thus W.n <> v by A17, A18, GLIB_001:87;
      A20: n+2 <= len W by A18, GLIB_001:1;
      assume W.(n+2) = v;
      hence contradiction by A17,A20, GLIB_001:87;
    end;
    then e Joins W.n,W.(n+2),G2 by A19, A1, Th49;
    hence thesis by GLIB_000:def 13;
  end;
  hence thesis by TARSKI:def 3;
end;
