
theorem Th2:
  for G being _Graph
  for V being non empty one-to-one ManySortedSet of the_Vertices_of G
  holds the_Vertices_of replaceVertices(V) = rng V &
    the_Edges_of replaceVertices(V) = the_Edges_of G &
    the_Source_of replaceVertices(V) = V*(the_Source_of G) &
    the_Target_of replaceVertices(V) = V*(the_Target_of G)
proof
  let G be _Graph;
  let V be non empty one-to-one ManySortedSet of the_Vertices_of G;
  thus the_Vertices_of replaceVertices(V) = rng V by Th1;
  rng id the_Edges_of G = the_Edges_of G;
  hence the_Edges_of replaceVertices(V) = the_Edges_of G by Th1;
  A1: (id the_Edges_of G)" = id the_Edges_of G by FUNCT_1:45;
  dom(V*(the_Source_of G)) c= the_Edges_of G;
  then V*(the_Source_of G) = V*(the_Source_of G)*((id the_Edges_of G)")
    by A1, RELAT_1:51;
  hence the_Source_of replaceVertices(V) = V*(the_Source_of G) by Th1;
  dom(V*(the_Target_of G)) c= the_Edges_of G;
  then V*(the_Target_of G) = V*(the_Target_of G)*((id the_Edges_of G)")
    by A1, RELAT_1:51;
  hence the_Target_of replaceVertices(V) = V*(the_Target_of G) by Th1;
end;
