
theorem Th30:
  for G1 being non locally-finite _Graph
  for V being finite Subset of the_Vertices_of G1
  for G2 being removeVertices of G1, V
  st for v being Vertex of G1 st v in V holds v.edgesInOut() is finite
  holds G2 is non locally-finite
proof
  let G1 be non locally-finite _Graph;
  let V be finite Subset of the_Vertices_of G1;
  let G2 be removeVertices of G1, V;
  assume A1: for v being Vertex of G1 st v in V holds v.edgesInOut() is finite;
  per cases;
  suppose A2: the_Vertices_of G1 \ V is non empty Subset of the_Vertices_of G1;
    consider v1 being Vertex of G1 such that
      A3: v1.edgesInOut() is infinite by Def5;
    A4: the_Vertices_of G2 = the_Vertices_of G1 \ V by A2, GLIB_000:def 37;
    not v1 in V by A1, A3;
    then reconsider v2 = v1 as Vertex of G2 by A4, XBOOLE_0:def 5;
    not the_Vertices_of G1 c= V by A2, XBOOLE_1:37;
    then V c< the_Vertices_of G1 by XBOOLE_0:def 8;
    then A5: v2.edgesInOut() = v1.edgesInOut() \ G1.edgesInOut(V)
      by GLIB_000:166;
    deffunc F(Vertex of G1) = $1.edgesInOut();
    set S = {F(v) where v is Vertex of G1 : v in V};
    A6: V is finite;
    A7: S is finite from FRAENKEL:sch 21(A6);
    now
      let X be set;
      assume X in S;
      then consider v being Vertex of G1 such that
        A8: X = F(v) & v in V;
      thus X is finite by A1, A8;
    end;
    then union S is finite by A7, FINSET_1:7;
    then G1.edgesInOut(V) is finite by GLIB_000:160;
    hence thesis by A3, A5;
  end;
  suppose not(the_Vertices_of G1\V is non empty Subset of the_Vertices_of G1);
    then G1 == G2 by GLIB_000:def 37;
    hence thesis by Th24;
  end;
end;
