
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 in dom E & v in dom V & w in dom V &
    E.e Joins V.v,V.w,replaceVerticesEdges(V,E)
  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 be one-to-one ManySortedSet of the_Edges_of G;
  let e,v,w be object;
  assume A1: e in dom E & v in dom V & w in dom V;
  assume E.e Joins V.v,V.w,replaceVerticesEdges(V,E);
  then per cases by GLIB_000:16;
  suppose E.e DJoins V.v,V.w,replaceVerticesEdges(V,E);
    then e DJoins v,w,G by A1, Th10;
    hence thesis by GLIB_000:16;
  end;
  suppose E.e DJoins V.w,V.v,replaceVerticesEdges(V,E);
    then e DJoins w,v,G by A1, Th10;
    hence thesis by GLIB_000:16;
  end;
end;
