
theorem
  for G2 being _Graph, V being Subset of the_Vertices_of G2
  for G1 being addLoops of G2, V for v being Vertex of G1
  st v in V holds v,v are_adjacent
proof
  let G2 be _Graph, V be Subset of the_Vertices_of G2, G1 be addLoops of G2, V;
  let v be Vertex of G1;
  consider E being set, f being one-to-one Function such that
    A1: E misses the_Edges_of G2 & the_Edges_of G1 = the_Edges_of G2 \/ E &
      dom f = E & rng f = V & the_Source_of G1 = the_Source_of G2 +* f &
      the_Target_of G1 = the_Target_of G2 +* f by Def5;
  assume v in V;
  then consider e being object such that
    A2: e in dom f & f.e = v by A1, FUNCT_1:def 3;
  A3: (the_Source_of G1).e = v by A1, A2, FUNCT_4:13;
  A4: (the_Target_of G1).e = v by A1, A2, FUNCT_4:13;
  e in the_Edges_of G1 by A1, A2, XBOOLE_0:def 3;
  then e Joins v,v,G1 by A3, A4, GLIB_000:def 13;
  hence v,v are_adjacent by CHORD:def 3;
end;
