
theorem Th3:
  for G being _Graph
  for E being one-to-one ManySortedSet of the_Edges_of G
  holds the_Vertices_of replaceEdges(E) = the_Vertices_of G &
    the_Edges_of replaceEdges(E) = rng E &
    the_Source_of replaceEdges(E) = (the_Source_of G)*(E") &
    the_Target_of replaceEdges(E) = (the_Target_of G)*(E")
proof
  let G be _Graph;
  let E be one-to-one ManySortedSet of the_Edges_of G;
  rng id the_Vertices_of G = the_Vertices_of G;
  hence the_Vertices_of replaceEdges(E) = the_Vertices_of G by Th1;
  thus the_Edges_of replaceEdges(E) = rng E by Th1;
  rng((the_Source_of G)*(E")) c= the_Vertices_of G;
  then (the_Source_of G)*(E")
     = (id the_Vertices_of G)*((the_Source_of G)*(E")) by RELAT_1:53
    .= (id the_Vertices_of G)*(the_Source_of G)*(E") by RELAT_1:36;
  hence the_Source_of replaceEdges(E) = (the_Source_of G)*(E") by Th1;
  rng((the_Target_of G)*(E")) c= the_Vertices_of G;
  then (the_Target_of G)*(E")
     = (id the_Vertices_of G)*((the_Target_of G)*(E")) by RELAT_1:53
    .= (id the_Vertices_of G)*(the_Target_of G)*(E") by RELAT_1:36;
  hence the_Target_of replaceEdges(E) = (the_Target_of G)*(E") by Th1;
end;
