
theorem
  for G1, G2 being _Graph, F being PGraphMapping of G1, G2
  st F is isomorphism holds G1 is locally-finite iff G2 is locally-finite
proof
  let G1, G2 be _Graph, F be PGraphMapping of G1, G2;
  assume A1: F is isomorphism;
  hereby
    assume A2: G1 is locally-finite;
    now
      let v be Vertex of G2;
      rng F_V = the_Vertices_of G2 by A1, GLIB_010:def 12;
      then consider v0 being object such that
        A3: v0 in dom F_V & F_V.v0 = v by FUNCT_1:def 3;
      reconsider v0 as Vertex of G1 by A3;
      (F_V/.v0).degree() = v0.degree() by A1, GLIBPRE0:93;
      hence v.degree() is finite by A2, A3, PARTFUN1:def 6;
    end;
    hence G2 is locally-finite by Th23;
  end;
  thus thesis by A1, Th32;
end;
