
theorem Th11:
  for G being _Graph
  for V being non empty one-to-one ManySortedSet of the_Vertices_of G
  for e,v,w being object
  st v in dom V & w in dom V & e DJoins V.v,V.w,replaceVertices(V)
  holds e DJoins v,w,G
proof
  let G be _Graph;
  let V be non empty one-to-one ManySortedSet of the_Vertices_of G;
  let e,v,w be object;
  assume A1: v in dom V & w in dom V & e DJoins V.v,V.w,replaceVertices(V);
  then e in the_Edges_of replaceVertices(V) by GLIB_000:def 14;
  then A2: e in the_Edges_of G by Th2;
  then A3: e in dom id the_Edges_of G;
  (id the_Edges_of G).e = e by A2, FUNCT_1:18;
  hence e DJoins v,w,G by A1, A3, Th10;
end;
