
theorem Th75:
  for G1 being complete _Graph, G2 being LGraphComplement of G1
  holds the_Edges_of G2 = G2.loops()
proof
  let G1 be complete _Graph, G2 be LGraphComplement of G1;
  now
    let e be object;
    set v = (the_Source_of G2).e, w = (the_Target_of G2).e;
    assume e in the_Edges_of G2;
    then A1: e Joins v,w,G2 by GLIB_000:def 13;
    v = w
    proof
      assume A2: v <> w;
      v is Vertex of G2 & w is Vertex of G2 by A1, GLIB_000:13;
      then reconsider v0=v, w0=w as Vertex of G1 by Def7;
      ex e0 being object st e0 Joins v0,w0,G1 by A2, CHORD:def 3, CHORD:def 6;
      hence contradiction by A1, Th64;
    end;
    hence e in G2.loops() by A1, GLIB_009:def 2;
  end;
  then the_Edges_of G2 c= G2.loops() by TARSKI:def 3;
  hence thesis by XBOOLE_0:def 10;
end;
