
theorem
  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 Joins V.v,V.w,replaceVertices(V)
  holds e Joins 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;
  assume e Joins V.v,V.w,replaceVertices(V);
  then per cases by GLIB_000:16;
  suppose e DJoins V.v,V.w,replaceVertices(V);
    then e DJoins v,w,G by A1, Th11;
    hence thesis by GLIB_000:16;
  end;
  suppose e DJoins V.w,V.v,replaceVertices(V);
    then e DJoins w,v,G by A1, Th11;
    hence thesis by GLIB_000:16;
  end;
end;
