
theorem
  for G1 being non _trivial _Graph, G2 being DGraphComplement of G1
  for v1 being Vertex of G1, v2 being Vertex of G2 st v1 = v2
  holds v1 is isolated implies v2 is non isolated
proof
  let G1 be non _trivial _Graph, G2 be DGraphComplement of G1;
  let v1 be Vertex of G1, v2 be Vertex of G2;
  assume A1: v1 = v2;
  assume A2: v1 is isolated;
  set v9 = the Element of the_Vertices_of G1 \ {v1};
  the_Vertices_of G1 \ {v1} <> {} by GLIB_000:20;
  then A3: v9 in the_Vertices_of G1 \ {v1};
  then reconsider v9 as Vertex of G1;
  A4: v9 <> v1 by A3, ZFMISC_1:56;
  not ex e being object st e DJoins v1,v9,G1 by A2, GLIB_000:144;
  then ex e being object st e DJoins v1,v9,G2 by A4, Th80;
  hence v2 is non isolated by A1, GLIB_000:144;
end;
