
theorem
  for G1 being _Graph, G2 being LGraphComplement 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) &
    (v1 is endvertex implies v2 is non endvertex)
proof
  let G1 be _Graph, G2 be LGraphComplement of G1;
  let v1 be Vertex of G1, v2 be Vertex of G2;
  assume A1: v1 = v2;
  hereby
    assume v1 is isolated;
    then not ex e1 being object st e1 Joins v1,v1,G1 by GLIB_000:143;
    then consider e2 being object such that
      A2: e2 Joins v1,v1,G2 by Def7;
    thus v2 is non isolated by A1, A2, GLIB_000:143;
  end;
  hereby
    assume v1 is endvertex;
    then not ex e1 being object st e1 Joins v1,v1,G1 by GLIB_000:146;
    then consider e2 being object such that
      A3: e2 Joins v1,v1,G2 by Def7;
    thus v2 is non endvertex by A1, A3, GLIB_000:146;
  end;
end;
