reserve G, G1, G2 for _Graph, H for Subgraph of G;

theorem Th15:
  for G being non edgeless _Graph, e being Edge of G, e0,v,w being object
  st e0 DJoins v,w,createGraph(e)
  holds e0 = e & v = (the_Source_of G).e & w = (the_Target_of G).e
proof
  let G be non edgeless _Graph, e be Edge of G, e0,v,w be object;
  assume A1: e0 DJoins v,w,createGraph(e);
  then e0 in the_Edges_of createGraph(e) by GLIB_000:def 14;
  then e0 in {e} by Th13;
  then A2: e0 = e & v is set & w is set by TARSKI:def 1, TARSKI:1;
  then e DJoins v,w,G by A1, GLIB_000:72;
  hence thesis by A2, GLIB_000:def 14;
end;
