
theorem Th29:
  for G2 being _Graph, v being object, V being Subset of the_Vertices_of G2
  for G1 being addAdjVertexAll of G2, v, V st not v in the_Vertices_of G2
  holds (G2 is locally-finite & V is finite) iff G1 is locally-finite
proof
  let G2 be _Graph, v be object, V be Subset of the_Vertices_of G2;
  let G1 be addAdjVertexAll of G2, v, V;
  assume A1: not v in the_Vertices_of G2;
  hereby
    assume A2: G2 is locally-finite & V is finite;
    now
      let v1 be Vertex of G1;
      per cases;
      suppose v1 = v;
        then v1.degree() = card V by A1, GLIBPRE0:55;
        hence v1.degree() is finite by A2;
      end;
      suppose v1 <> v;
        then A3: not v1 in {v} by TARSKI:def 1;
        the_Vertices_of G1 = the_Vertices_of G2 \/ {v} by A1, GLIB_007:def 4;
        then reconsider v2 = v1 as Vertex of G2 by A3, XBOOLE_0:def 3;
        per cases;
        suppose not v2 in V;
          then v1.degree() = v2.degree() by GLIBPRE0:56;
          hence v1.degree() is finite by A2;
        end;
        suppose v2 in V;
          then v1.degree() = v2.degree() +` 1 by A1, GLIBPRE0:57;
          hence v1.degree() is finite by A2;
        end;
      end;
    end;
    hence G1 is locally-finite by Th23;
  end;
  assume A4: G1 is locally-finite;
  G2 is Subgraph of G1 by GLIB_006:57;
  hence G2 is locally-finite by A4;
  reconsider w = v as Vertex of G1 by A1, GLIB_007:50;
  w.degree() is finite by A4;
  then card V is finite by A1, GLIBPRE0:55;
  hence V is finite;
end;
