reserve c,c1,c2 for Cardinal, G,G1,G2 for _Graph, v for Vertex of G;

theorem Th24:
  G is edgeless iff G is 0-regular
proof
  thus G is edgeless implies G is 0-regular;
  assume A1: G is 0-regular;
  assume G is non edgeless;
  then consider e being object such that
    A2: e in the_Edges_of G by XBOOLE_0:def 1;
  A3: e Joins (the_Source_of G).e,(the_Target_of G).e,G by A2, GLIB_000:def 13;
  then reconsider v = (the_Source_of G).e as Vertex of G by GLIB_000:13;
  v.degree() <> 0 by A3, GLIB_000:143, GLIB_000:157;
  hence contradiction by A1;
end;
