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

theorem Th145:
  for G2 for v1 being Vertex of G2, e, v2 being object
  for G1 being addAdjVertex of G2,v1,e,v2
  for w being Vertex of G1
  st not e in the_Edges_of G2 & not v2 in the_Vertices_of G2 & w = v2
  holds w is endvertex
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 Vertex of G1;
  assume that
    A1: not e in the_Edges_of G2 & not v2 in the_Vertices_of G2 and
    A2: w = v2;
  ex e1 being object st w.edgesInOut() = {e1} & not e1 Joins w,w,G1
  proof
    take e;
    for e1 being object holds e1 in w.edgesInOut() iff e1 = e
    proof
      let e1 be object;
      hereby
        assume e1 in w.edgesInOut();
        then A3: e1 in the_Edges_of G1 & ((the_Source_of G1).e1 = w or
          (the_Target_of G1).e1 = w) by GLIB_000:61;
        thus e1 = e
        proof
          assume A4: e1 <> e;
          per cases by A3;
          suppose (the_Source_of G1).e1 = w;
            then e1 Joins (the_Target_of G1).e1, w,G1 by A3, GLIB_000:def 13;
            hence contradiction by A1, A2, A4, Th137;
          end;
          suppose (the_Target_of G1).e1 = w;
            then e1 Joins (the_Source_of G1).e1, w,G1 by A3, GLIB_000:def 13;
            hence contradiction by A1, A2, A4, Th137;
          end;
        end;
      end;
      assume A5: e1 = e;
      e DJoins v1,v2,G1 by A1, Th135;
      then e in the_Edges_of G1 & (the_Target_of G1).e = w
        by A2, GLIB_000:def 14;
      hence thesis by A5, GLIB_000:61;
    end;
    hence w.edgesInOut() = {e} by TARSKI:def 1;
    w <> v1 by A1, A2;
    hence not e Joins w,w,G1 by A1, A2, Th137;
  end;
  hence thesis by GLIB_000:def 51;
end;
