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

theorem Th135:
  for G2 for v1 being Vertex of G2, e,v2 being object
  for G1 being addAdjVertex of G2,v1,e,v2
  st not v2 in the_Vertices_of G2 & not e in the_Edges_of G2
  holds e DJoins v1,v2,G1 & e Joins v1,v2,G1
proof
  let G2 be _Graph;
  let v1 be Vertex of G2;
  let e, v2 be object;
  let G1 be addAdjVertex of G2,v1,e,v2;
  assume A1: not v2 in the_Vertices_of G2 & not e in the_Edges_of G2;
  e in {e} by TARSKI:def 1;
  then e in the_Edges_of G2 \/ {e} by XBOOLE_0:def 3;
  then A3: e in the_Edges_of G1 by A1, Def13;
  ((the_Source_of G2) +* (e .--> v1)).e = v1 &
    ((the_Target_of G2) +* (e .--> v2)).e = v2 by FUNCT_4:113;
  then (the_Source_of G1).e = v1 & (the_Target_of G1).e = v2 by A1, Def13;
  hence e DJoins v1,v2,G1 by A3, GLIB_000:def 14;
  hence e Joins v1,v2,G1 by GLIB_000:16;
end;
