
theorem Th1:
  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
  holds the_Vertices_of replaceVerticesEdges(V,E) = rng V &
    the_Edges_of replaceVerticesEdges(V,E) = rng E &
    the_Source_of replaceVerticesEdges(V,E) = V*(the_Source_of G)*(E") &
    the_Target_of replaceVerticesEdges(V,E) = V*(the_Target_of G)*(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;
  consider S,T being Function of rng E, rng V such that
    A1: S = V*(the_Source_of G)*(E") & T = V*(the_Target_of G)*(E") and
    A2: replaceVerticesEdges(V,E) = createGraph(rng V,rng E,S,T) by Def1;
  thus thesis by A1, A2;
end;
