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

theorem Th67:
  for G, v, V for G1, G2 being addAdjVertexAll of G,v,V
  for v1,e1,v2 being object st e1 Joins v1,v2,G1
  ex e2 being object st e2 Joins v1,v2,G2
proof
  let G,v,V;
  let G1,G2 be addAdjVertexAll of G,v,V;
  let v1,e1,v2 be object;
  assume A1: e1 Joins v1,v2,G1;
  per cases;
  suppose A2: V c= the_Vertices_of G & not v in the_Vertices_of G;
    then consider E2 being set such that
      card V = card E2 & E2 misses the_Edges_of G &
        the_Edges_of G2 = the_Edges_of G \/ E2 and
      A3: for w1 being object st w1 in V ex e3 being object st e3 in E2 &
        e3 Joins w1,v,G2 &
        for e2 being object st e2 Joins w1,v,G2 holds e3 = e2
      by Def4;
    per cases;
    suppose v1 = v & v2 = v;
      hence thesis by A2, Def4, A1;
    end;
    suppose A4: v1 = v & v2 <> v;
      per cases;
      suppose not v2 in V;
        then not e1 Joins v2,v,G1 by A2, Def4;
        hence thesis by A1,A4, GLIB_000:14;
      end;
      suppose v2 in V;
        then consider e3 being object such that
          A5: e3 in E2 & e3 Joins v2,v,G2 and
          for e2 being object st e2 Joins v2,v,G2 holds e3 = e2 by A3;
        take e3;
        thus e3 Joins v1,v2,G2 by A5, A4, GLIB_000:14;
      end;
    end;
    suppose A6: v1 <> v & v2 = v;
      per cases;
      suppose not v1 in V;
        hence thesis by A1,A2, A6, Def4;
      end;
      suppose v1 in V;
        then consider e3 being object such that
          A7: e3 in E2 & e3 Joins v1,v,G2 and
          for e2 being object st e2 Joins v1,v,G2 holds e3 = e2 by A3;
        take e3;
        thus e3 Joins v1,v2,G2 by A7, A6;
      end;
    end;
    suppose A8: v1 <> v & v2 <> v;
      A9: e1 Joins v1,v2,G
      proof
        e1 DJoins v1,v2,G1 or e1 DJoins v2,v1,G1 by A1, GLIB_000:16;
        then e1 DJoins v1,v2,G or e1 DJoins v2,v1,G by A2, A8, Def4;
        hence thesis by GLIB_000:16;
      end;
      take e1;
      reconsider w1=v1, w2=v2 as set by TARSKI:1;
      e1 Joins w1,w2,G2 by A9, GLIB_006:70;
      hence thesis;
    end;
  end;
  suppose not (V c= the_Vertices_of G & not v in the_Vertices_of G);
    then G1 == G & G2 == G by Def4;
    then A10: G1 == G2 by GLIB_000:85;
    take e1;
    thus thesis by A1, A10, GLIB_000:88;
  end;
end;
