reserve G for _Graph;
reserve G2 for _Graph, G1 for Supergraph of G2;
reserve V for set;
reserve v for object;

theorem
  for G2 for v1, v2 being Vertex of G2, e being object
  for G1 being addEdge of G2,v1,e,v2 st not e in the_Edges_of G2
  for e1,w1,w2 being object
  holds e1 Joins w1,w2,G1 & not e1 in the_Edges_of G2
    implies (w1 = v1 & w2 = v2) or (w1 = v2 & w2 = v1)
proof
  let G2;
  let v1,v2 be Vertex of G2, e be object, G1 be addEdge of G2,v1,e,v2;
  assume A1: not e in the_Edges_of G2;
  let e1,w1,w2 be object;
  assume A2: e1 Joins w1,w2,G1 & not e1 in the_Edges_of G2;
  then A3: e = e1 by A1, Th110;
  e DJoins v1,v2,G1 by A1, Th109;
  then e1 Joins v1,v2,G1 by A3, GLIB_000:16;
  hence thesis by A2, GLIB_000:15;
end;
