
theorem Th3:
  for G1 being _Graph, E being set, G2 being reverseEdgeDirections of G1, E
  holds G1 is loopfull iff G2 is loopfull
proof
  let G1 be _Graph, E be set, G2 be reverseEdgeDirections of G1, E;
  hereby
    assume A1: G1 is loopfull;
    now
      let v be Vertex of G2;
      v is Vertex of G1 by GLIB_007:4;
      then consider e being object such that
        A2: e Joins v,v,G1 by A1;
      take e;
      thus e Joins v,v,G2 by A2, GLIB_007:9;
    end;
    hence G2 is loopfull;
  end;
  assume A3: G2 is loopfull;
  now
    let v be Vertex of G1;
    v is Vertex of G2 by GLIB_007:4;
    then consider e being object such that
      A4: e Joins v,v,G2 by A3;
    take e;
    thus e Joins v,v,G1 by A4, GLIB_007:9;
  end;
  hence G1 is loopfull;
end;
