reserve G, G2 for _Graph, V, E for set,
  v for object;

theorem Th53:
  for G2, v, V for G1 being addAdjVertexAll of G2,v,V
  st V c= the_Vertices_of G2 & not v in the_Vertices_of G2
  holds the_Edges_of G2 = G1.edgesBetween(the_Vertices_of G2)
proof
  let G2, v, V;
  let G1 be addAdjVertexAll of G2,v,V;
  assume A1: V c= the_Vertices_of G2 & not v in the_Vertices_of G2;
  set B = G1.edgesBetween(the_Vertices_of G2);
  for e being object holds e in the_Edges_of G2 iff e in B
  proof
    let e be object;
    hereby
      assume A2: e in the_Edges_of G2;
      then (the_Source_of G2).e in the_Vertices_of G2 &
        (the_Target_of G2).e in the_Vertices_of G2 by FUNCT_2:5;
      then A3: (the_Source_of G1).e in the_Vertices_of G2 &
        (the_Target_of G1).e in the_Vertices_of G2 by A2, GLIB_006:def 9;
      the_Edges_of G2 c= the_Edges_of G1 by GLIB_006:def 9;
      hence e in B by A3, GLIB_000:31, A2;
    end;
    set x = (the_Source_of G1).e, y = (the_Target_of G1).e;
    assume e in B;
    then A4: e in the_Edges_of G1 &
      x in the_Vertices_of G2 & y in the_Vertices_of G2 by GLIB_000:31; then
    e Joins x,y,G1 by GLIB_000:def 13;
    then e Joins x,y,G2 by A1, A4, Th49;
    hence thesis by GLIB_000:def 13;
  end;
  hence thesis by TARSKI:2;
end;
