
theorem
  for G2 being _Graph, v,w,e being object, G1 being addEdge of G2,v,e,w
  for v1 being Vertex of G1, v2 being Vertex of G2
  st v1 = v2 & v2 <> v & v2 <> w 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,w,e be object, G1 be addEdge of G2,v,e,w;
  let v1 be Vertex of G1, v2 be Vertex of G2;
  assume A1: v1 = v2 & v2 <> v & v2 <> w;
  per cases;
  suppose A2: v in the_Vertices_of G2 & w in the_Vertices_of G2 &
      not e in the_Edges_of G2;
    A3: G2 is Subgraph of G1 by GLIB_006:57;
    A4: v2.edgesIn() c= v1.edgesIn() by A1, A3, GLIB_000:78;
    now
      let f be object;
      assume f in v1.edgesIn();
      then consider x being set such that
        A5: f DJoins x,v1,G1 by GLIB_000:57;
      f in the_Edges_of G2
      proof
        assume A7: not f in the_Edges_of G2;
        f Joins x,v1,G1 by A5, GLIB_000:16;
        hence contradiction by A1, A2, A7, GLIB_006:107;
      end;
      hence f in v2.edgesIn() by A1, A5, GLIB_006:71, GLIB_000:57;
    end;
    then v1.edgesIn() c= v2.edgesIn() by TARSKI:def 3;
    hence A8: v1.edgesIn() = v2.edgesIn() by A4, XBOOLE_0:def 10;
    hence v1.inDegree() = v2.inDegree();
    A9: v2.edgesOut() c= v1.edgesOut() by A1, A3, GLIB_000:78;
    now
      let f be object;
      assume f in v1.edgesOut();
      then consider x being set such that
        A10: f DJoins v1,x,G1 by GLIB_000:59;
      f in the_Edges_of G2
      proof
        assume A12: not f in the_Edges_of G2;
        f Joins x,v1,G1 by A10, GLIB_000:16;
        hence contradiction by A1, A2, A12, GLIB_006:107;
      end;
      hence f in v2.edgesOut() by A1, A10, GLIB_006:71, GLIB_000:59;
    end;
    then v1.edgesOut() c= v2.edgesOut() by TARSKI:def 3;
    hence A13: v1.edgesOut() = v2.edgesOut() by A9, XBOOLE_0:def 10;
    hence v1.outDegree() = v2.outDegree();
    thus thesis by A8, A13;
  end;
  suppose 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 11;
    hence thesis by A1, GLIB_000:96;
  end;
end;
