
theorem Th10:
  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 DJoins V.v,V.w,replaceVerticesEdges(V,E)
  holds e DJoins 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;
  then A2: e in the_Edges_of G;
  assume A3: E.e DJoins V.v,V.w,replaceVerticesEdges(V,E);
  E.e in rng E by A1, FUNCT_1:3;
  then A4: E.e in dom(E") by FUNCT_1:33;
  A5: e in dom the_Source_of G & e in dom the_Target_of G by A2, FUNCT_2:def 1;
  (the_Source_of G).e in the_Vertices_of G &
    (the_Target_of G).e in the_Vertices_of G by A1, FUNCT_2:5;
  then A6: (the_Source_of G).e in dom V &
    (the_Target_of G).e in dom V by PARTFUN1:def 2;
  V.v = (the_Source_of replaceVerticesEdges(V,E)).(E.e) by A3, GLIB_000:def 14
    .= (V*(the_Source_of G)*(E")).(E.e) by Th1
    .= (V*(the_Source_of G)).(E".(E.e)) by A4, FUNCT_1:13
    .= (V*(the_Source_of G)).e by A1, FUNCT_1:34
    .= V.((the_Source_of G).e) by A5, FUNCT_1:13;
  then A7: v = (the_Source_of G).e by A1, A6, FUNCT_1:def 4;
  V.w = (the_Target_of replaceVerticesEdges(V,E)).(E.e) by A3, GLIB_000:def 14
    .= (V*(the_Target_of G)*(E")).(E.e) by Th1
    .= (V*(the_Target_of G)).(E".(E.e)) by A4, FUNCT_1:13
    .= (V*(the_Target_of G)).e by A1, FUNCT_1:34
    .= V.((the_Target_of G).e) by A5, FUNCT_1:13;
  then w = (the_Target_of G).e by A1, A6, FUNCT_1:def 4;
  hence e DJoins v,w,G by A1, A7, GLIB_000:def 14;
end;
