
theorem Th10:
  for G2 being _Graph, v being Vertex of G2, e,w being object
  for G1 being addAdjVertex of G2,v,e,w for v1 being Vertex of G1
  st not e in the_Edges_of G2 & not w in the_Vertices_of G2 & v1 = v holds
    v1.edgesIn() = v.edgesIn() & v1.inDegree() = v.inDegree() &
    v1.edgesOut() = v.edgesOut() \/ {e} & v1.outDegree() = v.outDegree() +` 1 &
    v1.edgesInOut() = v.edgesInOut() \/ {e} & v1.degree() = v.degree() +` 1
proof
  let G2 be _Graph, v being Vertex of G2, e,w be object;
  let G1 be addAdjVertex of G2,v,e,w, v1 be Vertex of G1;
  assume A1: not e in the_Edges_of G2 & not w in the_Vertices_of G2 & v1 = v;
  then consider G3 being addVertex of G2,w such that
    A2: G1 is addEdge of G3,v,e,w by GLIB_006:125;
  reconsider v3 = v as Vertex of G3 by GLIB_006:68;
  A3: v3.edgesIn() = v.edgesIn() & v3.inDegree() = v.inDegree() &
    v3.edgesOut() = v.edgesOut() & v3.outDegree() = v.outDegree() &
    v3.edgesInOut() = v.edgesInOut() & v3.degree() = v.degree()
    by GLIBPRE0:45;
  A4: the_Edges_of G3 = the_Edges_of G2 by GLIB_006:def 10;
  w in {w} by TARSKI:def 1;
  then A5: w is Vertex of G3 by GLIB_006:86;
  v <> w by A1;
  hence thesis by A1, A2, A3, A4, A5, GLIBPRE0:47;
end;
