
theorem Th12:
  for G being _Graph, E being one-to-one ManySortedSet of the_Edges_of G
  for e,v,w being object st e in dom E & E.e DJoins v,w,replaceEdges(E)
  holds e DJoins v,w,G
proof
  let G be _Graph;
  let E be one-to-one ManySortedSet of the_Edges_of G;
  let e,v,w be object;
  assume A1: e in dom E & E.e DJoins v,w,replaceEdges(E);
  then E.e Joins v,w,replaceEdges(E) by GLIB_000:16;
  then v in the_Vertices_of replaceEdges(E) &
    w in the_Vertices_of replaceEdges(E) by GLIB_000:13;
  then A2: v in the_Vertices_of G & w in the_Vertices_of G by Th3;
  then A3: v in dom id the_Vertices_of G & w in dom id the_Vertices_of G;
  (id the_Vertices_of G).v = v & (id the_Vertices_of G).w = w
    by A2, FUNCT_1:18;
  hence e DJoins v,w,G by A1, A3, Th10;
end;
