reserve G for _Graph;
reserve G2 for _Graph, G1 for Supergraph of G2;
reserve V for set;
reserve v for object;

theorem Th149:
  for G2 for v1 being Vertex of G2, e,v2 being object
  for G1 being addAdjVertex of G2,v1,e,v2, W being Walk of G1
  st not e in the_Edges_of G2 & not v2 in the_Vertices_of G2 &
    ((not e in W.edges() & W is non trivial) or not v2 in W.vertices())
  holds W is Walk of G2
proof
  let G2;
  let v1 be Vertex of G2, e,v2 be object;
  let G1 be addAdjVertex of G2,v1,e,v2;
  let W be Walk of G1;
  assume that
    A1: not e in the_Edges_of G2 & not v2 in the_Vertices_of G2 and
    A2: (not e in W.edges() & W is non trivial) or not v2 in W.vertices();
  reconsider w=v2, e1=e as set by TARSKI:1;
  per cases by A2;
  suppose A3: not e in W.edges() & W is non trivial;
    consider G3 being addVertex of G2,v2 such that
      A4: G1 is addEdge of G3,v1,e,v2 by A1, Th129;
    A5: not e in the_Edges_of G3 by A1, Def10;
    A6: v1 is Vertex of G3 by Th72;
    v2 is Vertex of G3 by Th98;
    then G3 is removeEdge of G1, e1 by A4, A5, A6, Th112;
    then reconsider W3=W as Walk of G3 by A3, GLIB_001:172;
    W3.vertices() misses {v2} \ the_Vertices_of G2 by A3, Th94;
    hence thesis by Th95;
  end;
  suppose A7: not v2 in W.vertices();
    reconsider G3=G2 as removeVertex of G1, w by A1, Th141;
    reconsider w as Vertex of G1 by A1, Th133;
    A8: not w in W.vertices() by A7;
    G1 is non _trivial by A1, Th147;
    then W is Walk of G3 by A8, GLIB_001:171;
    hence thesis;
  end;
end;
