reserve G, G2 for _Graph, V, E for set,
  v for object;

theorem Th43:
  for G2, v, V for G1 being addAdjVertexFromAll of G2,v,V, v1 being object
  st V c= the_Vertices_of G2 & not v in the_Vertices_of G2 & v1 in V
  holds [v1,the_Edges_of G2] DJoins v1,v,G1
proof
  let G2, v, V;
  let G1 be addAdjVertexFromAll of G2,v,V, v1 be object;
  assume A1: V c= the_Vertices_of G2 & not v in the_Vertices_of G2 & v1 in V;
  then A2: the_Edges_of G1 = the_Edges_of G2 \/ (V --> the_Edges_of G2) &
    the_Source_of G1 = the_Source_of G2 +* pr1(V,{the_Edges_of G2}) &
    the_Target_of G1 = the_Target_of G2 +* ((V --> the_Edges_of G2) --> v)
    by Def3;
  set e = [v1,the_Edges_of G2];
  A3: the_Edges_of G2 in {the_Edges_of G2} by TARSKI:def 1;
  then A4: e in [: V, {the_Edges_of G2} :] by A1, ZFMISC_1:def 2;
  then A5: e in V --> the_Edges_of G2 by FUNCOP_1:def 2;
  then A6: e in dom ((V --> the_Edges_of G2) --> v);
  A7: e in dom pr1(V, {the_Edges_of G2}) by A4, FUNCT_3:def 4;
  A8: e in the_Edges_of G1 by A2, A5, XBOOLE_0:def 3;
  A9: (the_Target_of G1).e
     = ((V --> the_Edges_of G2) --> v).e by A2, A6, FUNCT_4:13
    .= v by A5, FUNCOP_1:7;
  (the_Source_of G1).e
     = pr1(V,{the_Edges_of G2}).e by A2, A7, FUNCT_4:13
    .= pr1(V,{the_Edges_of G2}).(v1, the_Edges_of G2) by BINOP_1:def 1
    .= v1 by A1, A3, FUNCT_3:def 4;
  hence thesis by A8, A9, GLIB_000:def 14;
end;
