
theorem
  for G being _Graph
  for V being non empty one-to-one ManySortedSet of the_Vertices_of G
  for E being one-to-one ManySortedSet of the_Edges_of G
  for e,v,w being object st e Joins v,w,G
  holds E.e Joins V.v,V.w,replaceVerticesEdges(V,E)
proof
  let G be _Graph;
  let V be non empty one-to-one ManySortedSet of the_Vertices_of G;
  let E be one-to-one ManySortedSet of the_Edges_of G;
  let e,v,w be object;
  assume e Joins v,w,G;
  then per cases by GLIB_000:16;
  suppose e DJoins v,w,G;
    then E.e DJoins V.v,V.w,replaceVerticesEdges(V,E) by Th4;
    hence thesis by GLIB_000:16;
  end;
  suppose e DJoins w,v,G;
    then E.e DJoins V.w,V.v,replaceVerticesEdges(V,E) by Th4;
    hence thesis by GLIB_000:16;
  end;
end;
