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

theorem Th33:
  for G2, v, V for G1 being addAdjVertexToAll of G2,v,V
  st V c= the_Vertices_of G2 & not v in the_Vertices_of G2
  holds G1.edgesOutOf({v}) = V --> the_Edges_of G2
proof
  let G2, v, V;
  let G1 be addAdjVertexToAll of G2,v,V;
  assume A1: V c= the_Vertices_of G2 & not v in the_Vertices_of G2;
  then A2: the_Edges_of G1 = the_Edges_of G2 \/ (V --> the_Edges_of G2) &
    the_Source_of G1 = the_Source_of G2 +* ((V --> the_Edges_of G2) --> v)
    by Def2;
  for e being object holds e in G1.edgesOutOf({v}) iff
    e in V --> the_Edges_of G2
  proof
    let e be object;
    reconsider e1=e as set by TARSKI:1;
    hereby
      assume e in G1.edgesOutOf({v});
      then A3: e1 in the_Edges_of G1 & (the_Source_of G1).e1 in {v}
        by GLIB_000:def 27;
      not e in the_Edges_of G2
      proof
        assume A4: e in the_Edges_of G2;
        then (the_Source_of G2).e1 = (the_Source_of G1).e1 by GLIB_006:def 9
          .= v by A3, TARSKI:def 1;
        hence contradiction by A1, A4, FUNCT_2:5;
      end;
      hence e in V --> the_Edges_of G2 by A2, A3, XBOOLE_0:def 3;
    end;
    assume A5: e in V --> the_Edges_of G2;
    then e in dom ((V --> the_Edges_of G2) --> v);
    then (the_Source_of G1).e
       = ((V --> the_Edges_of G2) --> v).e by A2, FUNCT_4:13
      .= v by A5, FUNCOP_1:7;
    then A6: (the_Source_of G1).e in {v} by TARSKI:def 1;
    e in the_Edges_of G1 by A5, A2, XBOOLE_0:def 3;
    hence thesis by A6, GLIB_000:def 27;
  end;
  hence thesis by TARSKI:2;
end;
