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

theorem Th34:
  for G2, v, V for G1 being addAdjVertexFromAll of G2,v,V
  st V c= the_Vertices_of G2 & not v in the_Vertices_of G2
  holds G1.edgesInto({v}) = V --> the_Edges_of G2
proof
  let G2, v, V;
  let G1 be addAdjVertexFromAll 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_Target_of G1 = the_Target_of G2 +* ((V --> the_Edges_of G2) --> v)
    by Def3;
  for e being object holds e in G1.edgesInto({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.edgesInto({v});
      then A3: e1 in the_Edges_of G1 & (the_Target_of G1).e1 in {v}
        by GLIB_000:def 26;
      then A4: (the_Target_of G1).e = v by TARSKI:def 1;
      not e in the_Edges_of G2
      proof
        assume A5: e in the_Edges_of G2;
        then (the_Target_of G2).e1 = v by A4, GLIB_006:def 9;
        hence contradiction by A1, A5, FUNCT_2:5;
      end;
      hence e in V --> the_Edges_of G2 by A2, A3, XBOOLE_0:def 3;
    end;
    assume A6: e in V --> the_Edges_of G2;
    then e in dom ((V --> the_Edges_of G2) --> v);
    then (the_Target_of G1).e
       = ((V --> the_Edges_of G2) --> v).e by A2, FUNCT_4:13
      .= v by A6, FUNCOP_1:7;
    then A7: (the_Target_of G1).e in {v} by TARSKI:def 1;
    e in the_Edges_of G1 by A6, A2, XBOOLE_0:def 3;
    hence thesis by A7, GLIB_000:def 26;
  end;
  hence thesis by TARSKI:2;
end;
