
theorem Th122:
  for G1 being _Graph, G2 being removeLoops of G1, G3 being DSimpleGraph of G1
  holds G3 is removeDParallelEdges of G2
proof
  let G1 be _Graph, G2 being removeLoops of G1, G3 be DSimpleGraph of G1;
  consider E being RepDEdgeSelection of G1 such that
    A1: G3 is inducedSubgraph of G1,the_Vertices_of G1,E\G1.loops() by Def10;
  A2: the_Vertices_of G1 c= the_Vertices_of G1 &
    the_Edges_of G1 = G1.edgesBetween(the_Vertices_of G1) by GLIB_000:34;
  then A3: the_Vertices_of G3 = the_Vertices_of G1 &
    the_Edges_of G3 = E\G1.loops() by A1, GLIB_000:def 37;
  A4: the_Vertices_of G2 = the_Vertices_of G1 &
    the_Edges_of G2 = the_Edges_of G1 \ G1.loops() by GLIB_000:53;
  set E2 = E \ G1.loops();
  A5: E2 c= the_Edges_of G2 by A4, XBOOLE_1:33;
  now
    let v,w,e0 be object;
    A6: v is set & w is set by TARSKI:1;
    assume A7: e0 DJoins v,w,G2;
    then A8: e0 DJoins v,w,G1 by A6, GLIB_000:72;
    then consider e being object such that
      A9: e DJoins v,w,G1 & e in E and
      A10: for e9 being object st e9 DJoins v,w,G1 & e9 in E holds e9 = e
      by Def6;
    take e;
    A11: not e in G1.loops()
    proof
      assume e in G1.loops();
      then consider u being object such that
        A12: e DJoins u,u,G1 by Th45;
      v = u & w = u by A9, A12, GLIB_000:125;
      then A13: e0 in G1.loops() by A8, Th45;
      e0 in the_Edges_of G2 by A7, GLIB_000:def 14;
      hence contradiction by A4, A13, XBOOLE_0:def 5;
    end;
    then e in the_Edges_of G2 by A4, A9, XBOOLE_0:def 5;
    hence e DJoins v,w,G2 by A6, A9, GLIB_000:73;
    thus e in E2 by A9, A11, XBOOLE_0:def 5;
    let e9 be object;
    assume e9 DJoins v,w,G2 & e9 in E2;
    then e9 DJoins v,w,G1 & e9 in E2 by A6, GLIB_000:72;
    then e9 DJoins v,w,G1 & e9 in E by XBOOLE_1:36, TARSKI:def 3;
    hence e9 = e by A10;
  end;
  then reconsider E2 as RepDEdgeSelection of G2 by A5, Def6;
  A14: the_Vertices_of G3 = the_Vertices_of G2 by A3, A4;
  E\G1.loops() c= the_Edges_of G1\G1.loops() by XBOOLE_1:33;
  then A15: G3 is Subgraph of G2 by A1, A2, GLIB_000:46;
  the_Vertices_of G2 c= the_Vertices_of G2 &
    the_Edges_of G2 = G2.edgesBetween(the_Vertices_of G2) by GLIB_000:34;
  then G3 is inducedSubgraph of G2, the_Vertices_of G2, E2
    by A3, A14, A15, GLIB_000:def 37;
  hence thesis by Def8;
end;
