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

theorem Th110:
  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 e1 = e
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: e1 in the_Edges_of G1 by GLIB_000:def 13;
  the_Edges_of G1 = the_Edges_of G2 \/ {e} by A1, Def11;
  then e1 in {e} by A3, A2, XBOOLE_0:def 3;
  hence e1 = e by TARSKI:def 1;
end;
