
theorem Th11:
  for G2 being _Graph, v, e being object, w being Vertex of G2
  for G1 being addAdjVertex of G2,v,e,w for w1 being Vertex of G1
  st not e in the_Edges_of G2 & not v in the_Vertices_of G2 & w1 = w holds
    w1.edgesIn() = w.edgesIn() \/ {e} & w1.inDegree() = w.inDegree() +` 1 &
    w1.edgesOut() = w.edgesOut() & w1.outDegree() = w.outDegree() &
    w1.edgesInOut() = w.edgesInOut() \/ {e} & w1.degree() = w.degree() +` 1
proof
  let G2 be _Graph, v, e be object, w be Vertex of G2;
  let G1 be addAdjVertex of G2,v,e,w, w1 be Vertex of G1;
  assume A1: not e in the_Edges_of G2 & not v in the_Vertices_of G2 & w1 = w;
  then consider G3 being addVertex of G2,v such that
    A2: G1 is addEdge of G3,v,e,w by GLIB_006:126;
  reconsider w3 = w as Vertex of G3 by GLIB_006:68;
  A3: w3.edgesIn() = w.edgesIn() & w3.inDegree() = w.inDegree() &
    w3.edgesOut() = w.edgesOut() & w3.outDegree() = w.outDegree() &
    w3.edgesInOut() = w.edgesInOut() & w3.degree() = w.degree()
    by GLIBPRE0:45;
  A4: the_Edges_of G3 = the_Edges_of G2 by GLIB_006:def 10;
  v in {v} by TARSKI:def 1;
  then A5: v is Vertex of G3 by GLIB_006:86;
  v <> w by A1;
  hence thesis by A1, A2, A3, A4, A5, GLIBPRE0:48;
end;
