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

theorem Th109:
  for G2 for v1, v2 being Vertex of G2, e being object
  for G1 being addEdge of G2,v1,e,v2
  holds not e in the_Edges_of G2 implies e DJoins v1,v2,G1
proof
  let G2;
  let v1, v2 be Vertex of G2;
  let e be object;
  let G1 be addEdge of G2,v1,e,v2;
  assume A1: 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 A2: e in the_Edges_of G1 by A1, Def11;
  A3: e in dom (e .--> v1) by FUNCOP_1:74;
  A4: (the_Source_of G1).e
     = (the_Source_of G2 +* (e .--> v1)).e by A1, Def11
    .= (e .--> v1).e by A3, FUNCT_4:13
    .= v1 by FUNCOP_1:72;
  A5: e in dom (e .--> v2) by FUNCOP_1:74;
  (the_Target_of G1).e
     = (the_Target_of G2 +* (e .--> v2)).e by A1, Def11
    .= (e .--> v2).e by A5, FUNCT_4:13
    .= v2 by FUNCOP_1:72;
  hence thesis by A2, A4, GLIB_000:def 14;
end;
