
theorem Th39:
  for G2 being _Graph, V being set, G1 being addLoops of G2, V
  holds G1 is _finite iff G2 is _finite
proof
  let G2 be _Graph, V be set, G1 be addLoops of G2, V;
  per cases;
  suppose A1: V c= the_Vertices_of G2;
    then consider E being set, f being one-to-one Function such that
      A2: E misses the_Edges_of G2 & the_Edges_of G1 = the_Edges_of G2 \/ E &
        dom f = E & rng f = V & the_Source_of G1 = the_Source_of G2 +* f &
        the_Target_of G1 = the_Target_of G2 +* f by Def5;
    thus G1 is _finite implies G2 is _finite;
    assume A3: G2 is _finite;
    then E is finite by A1, A2, CARD_1:59;
    then A4: the_Edges_of G1 is finite by A2, A3;
    the_Vertices_of G2 = the_Vertices_of G1 by Th15;
    hence thesis by A3, A4, GLIB_000:def 17;
  end;
  suppose not V c= the_Vertices_of G2;
    then G1 == G2 by Def5;
    hence thesis by GLIB_000:89;
  end;
end;
