
theorem Th73:
  for G1 being _Graph, G2 being LGraphComplement of G1 holds
    (G1 is _trivial iff G2 is _trivial) &
    (G1 is loopfull iff G2 is loopless) &
    (G1 is loopless iff G2 is loopfull)
proof
  let G1 be _Graph, G2 be LGraphComplement of G1;
  hereby
    assume G1 is _trivial;
    then 1 = G1.order() by GLIB_000:26
      .= G2.order() by Th72;
    hence G2 is _trivial by GLIB_000:26;
  end;
  hereby
    assume G2 is _trivial;
    then 1 = G2.order() by GLIB_000:26
      .= G1.order() by Th72;
    hence G1 is _trivial by GLIB_000:26;
  end;
  hereby
    assume A1: G1 is loopfull;
    now
      let v be object;
      given e2 being object such that
        A2: e2 Joins v,v,G2;
      v in the_Vertices_of G2 by A2, GLIB_000:13;
      then v in the_Vertices_of G1 by Def7;
      then consider e1 being object such that
        A3: e1 Joins v,v,G1 by A1;
      thus contradiction by A2, A3, Th64;
    end;
    hence G2 is loopless by GLIB_000:18;
  end;
  hereby
    assume A4: G2 is loopless;
    now
      let v be Vertex of G1;
      assume not ex e1 being object st e1 Joins v,v,G1;
      then ex e2 being object st e2 Joins v,v,G2 by Def7;
      hence contradiction by A4, GLIB_000:18;
    end;
    hence G1 is loopfull;
  end;
  hereby
    assume A5: G1 is loopless;
    now
      let v be Vertex of G2;
      A6: v is Vertex of G1 by Def7;
      assume not ex e2 being object st e2 Joins v,v,G2;
      then ex e1 being object st e1 Joins v,v,G1 by A6, Def7;
      hence contradiction by A5, GLIB_000:18;
    end;
    hence G2 is loopfull;
  end;
  hereby
    assume A7: G2 is loopfull;
    now
      let v be object;
      given e1 being object such that
        A8: e1 Joins v,v,G1;
      v in the_Vertices_of G1 by A8, GLIB_000:13;
      then v in the_Vertices_of G2 by Def7;
      then consider e2 being object such that
        A9: e2 Joins v,v,G2 by A7;
      thus contradiction by A8, A9, Th64;
    end;
    hence G1 is loopless by GLIB_000:18;
  end;
end;
