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

theorem Th58:
  for G2, v, V, E for G1 being addAdjVertexAll of G2,v,V
  st V c= the_Vertices_of G2 & not v in the_Vertices_of G2 &
    E misses the_Edges_of G2 & the_Edges_of G1 = the_Edges_of G2 \/ E
  holds E = G1.edgesBetween(V,{v})
proof
  let G2,v,V,E;
  let G1 be addAdjVertexAll of G2,v,V;
  assume that
    A1: V c= the_Vertices_of G2 & not v in the_Vertices_of G2 and
    A2: E misses the_Edges_of G2 & the_Edges_of G1 = the_Edges_of G2 \/ E;
  consider E1 being set such that
    A3: card V = card E1 & E1 misses the_Edges_of G2 and
    A4: the_Edges_of G1 = the_Edges_of G2 \/ E1 and
    A5: for v1 being object st v1 in V ex e1 being object st
      e1 in E1 & e1 Joins v1,v,G1 &
      for e2 being object st e2 Joins v1,v,G1 holds e1 = e2 by A1, Def4;
  A6: E = E1 by A2, A3, A4, XBOOLE_1:71;
  A7: E /\ the_Edges_of G2 = {} by A2, XBOOLE_0:def 7;
  for e being object holds e in E iff e in G1.edgesBetween(V,{v})
  proof
    let e be object;
    set x = (the_Source_of G1).e, y = (the_Target_of G1).e;
    hereby
      assume A8: e in E;
      then e in the_Edges_of G1 by A2, XBOOLE_0:def 3;
      then A9: e Joins x,y,G1 by GLIB_000:def 13;
      not e in the_Edges_of G2 by A7, A8, Lm7;
      then per cases by A1, A2, A9, Th51;
      suppose x = v & y in V;
        then x in {v} & y in V by TARSKI:def 1;
        then e SJoins V,{v},G1 by A9, GLIB_000:17;
        hence e in G1.edgesBetween(V,{v}) by GLIB_000:def 30;
      end;
      suppose y = v & x in V;
        then y in {v} & x in V by TARSKI:def 1;
        then e SJoins V,{v},G1 by A9, GLIB_000:17;
        hence e in G1.edgesBetween(V,{v}) by GLIB_000:def 30;
      end;
    end;
    assume e in G1.edgesBetween(V,{v});
    then A10: e SJoins V,{v},G1 by GLIB_000:def 30;
    then A11: e in the_Edges_of G1 by GLIB_000:def 15;
    per cases by A10, GLIB_000:def 15;
    suppose A12: x in V & y in {v};
      then consider e1 being object such that
        A13: e1 in E1 & e1 Joins x,v,G1 and
        A14: for e2 being object st e2 Joins x,v,G1 holds e1 = e2 by A5;
      y = v by A12, TARSKI:def 1;
      then e Joins x,v,G1 by A11, GLIB_000:def 13;
      hence e in E by A6, A13, A14;
    end;
    suppose A15: x in {v} & y in V;
      then consider e1 being object such that
        A16: e1 in E1 & e1 Joins y,v,G1 and
        A17: for e2 being object st e2 Joins y,v,G1 holds e1 = e2 by A5;
      x = v by A15, TARSKI:def 1;
      then e Joins y,v,G1 by A11, GLIB_000:def 13;
      hence e in E by A6, A16, A17;
    end;
  end;
  hence thesis by TARSKI:2;
end;
