
theorem
  for G being _Graph, H being removeLoops of G, H9 being loopless Subgraph of G
  holds H9 is Subgraph of H
proof
  let G be _Graph, H be removeLoops of G, H9 be loopless Subgraph of G;
  the_Vertices_of H = the_Vertices_of G by GLIB_000:53;
  then A1: the_Vertices_of H9 c= the_Vertices_of H;
  A2: the_Edges_of H = the_Edges_of G \ G.loops() by GLIB_000:53;
  the_Edges_of H9 /\ G.loops() = {}
  proof
    assume the_Edges_of H9 /\ G.loops() <> {};
    then consider x being object such that
      A3: x in the_Edges_of H9 /\ G.loops() by XBOOLE_0:def 1;
    A4: x in the_Edges_of H9 & x in G.loops() by A3, XBOOLE_0:def 4;
    then consider v being object such that
      A5: x Joins v,v,G by GLIB_009:def 2;
    x is set & v is set by TARSKI:1;
    then x Joins v,v,H9 by A4, A5, GLIB_000:73;
    hence contradiction by GLIB_000:18;
  end;
  then the_Edges_of H9 misses G.loops() by XBOOLE_0:def 7;
  hence thesis by A1, A2, XBOOLE_1:86, GLIB_000:44;
end;
