
theorem Th9:
  for G2 being _Graph, v,e,w being object, v2 being Vertex of G2
  for G1 being addAdjVertex of G2,v,e,w for v1 being Vertex of G1
  st v1 <> v & v1 <> w & v1 = v2 holds
    v1.edgesIn() = v2.edgesIn() & v1.inDegree() = v2.inDegree() &
    v1.edgesOut() = v2.edgesOut() & v1.outDegree() = v2.outDegree() &
    v1.edgesInOut() = v2.edgesInOut() & v1.degree() = v2.degree()
proof
  let G2 be _Graph, v,e,w be object, v2 be Vertex of G2;
  let G1 be addAdjVertex of G2,v,e,w, v1 be Vertex of G1;
  assume A1: v1 <> v & v1 <> w & v1 = v2;
  per cases;
  suppose v in the_Vertices_of G2 & not w in the_Vertices_of G2 &
      not e in the_Edges_of G2;
    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 = v2 as Vertex of G3 by GLIB_006:68;
    v3.edgesIn() = v2.edgesIn() & v3.inDegree() = v2.inDegree() &
      v3.edgesOut() = v2.edgesOut() & v3.outDegree() = v2.outDegree() &
      v3.edgesInOut() = v2.edgesInOut() & v3.degree() = v2.degree()
      by GLIBPRE0:45;
    hence thesis by A1, A2, GLIBPRE0:46;
  end;
  suppose not v in the_Vertices_of G2 & w in the_Vertices_of G2 &
      not e in the_Edges_of G2;
    then consider G3 being addVertex of G2,v such that
      A3: G1 is addEdge of G3,v,e,w by GLIB_006:126;
    reconsider v3 = v2 as Vertex of G3 by GLIB_006:68;
    v3.edgesIn() = v2.edgesIn() & v3.inDegree() = v2.inDegree() &
      v3.edgesOut() = v2.edgesOut() & v3.outDegree() = v2.outDegree() &
      v3.edgesInOut() = v2.edgesInOut() & v3.degree() = v2.degree()
      by GLIBPRE0:45;
    hence thesis by A1, A3, GLIBPRE0:46;
  end;
  suppose not((v in the_Vertices_of G2 & not w in the_Vertices_of G2 &
      not e in the_Edges_of G2)or(not v in the_Vertices_of G2 &
      w in the_Vertices_of G2 & not e in the_Edges_of G2));
    then G1 == G2 by GLIB_006:def 12;
    hence thesis by A1, GLIB_000:96;
  end;
end;
