
theorem
  for G2 being _Graph, V being set, G1 being addLoops of G2, V
  for W being Walk of G1 st W.edges() misses G1.loops() \ G2.loops()
  holds W is Walk of G2
proof
  let G2 be _Graph, V be set, G1 be addLoops of G2, V;
  let W be Walk of G1;
  A1: G2 is Subgraph of G1 by GLIB_006:57;
  assume A2: W.edges() misses G1.loops() \ G2.loops();
  per cases;
  suppose V c= the_Vertices_of G2;
    then consider E being set, f being one-to-one Function such that
      A3: E misses the_Edges_of G2 & the_Edges_of G1 = the_Edges_of G2 \/ E &
        dom f = E & rng f = V & the_Source_of G1 = the_Source_of G2 +* f &
        the_Target_of G1 = the_Target_of G2 +* f by Def5;
    now
      let e be object;
      assume A4: e in E;
      then A5: (the_Source_of G1).e = f.e & (the_Target_of G1).e = f.e
        by A3, FUNCT_4:13;
      A6: e in the_Edges_of G2 \/ E by A4, XBOOLE_0:def 3;
      then e Joins f.e,f.e,G1 by A3, A5, GLIB_000:def 13;
      then A7: e in G1.loops() by GLIB_009:def 2;
      not e in G2.loops() by A3, A4, A6, XBOOLE_0:5;
      hence e in G1.loops() \ G2.loops() by A7, XBOOLE_0:def 5;
    end;
    then W.edges() misses E by A2, XBOOLE_1:63, TARSKI:def 3;
    then A8: W.edges() c= the_Edges_of G2 by A3, XBOOLE_1:73;
    W.vertices() c= the_Vertices_of G1;
    then W.vertices() c= the_Vertices_of G2 by Th15;
    hence thesis by A1, A8, GLIB_001:170;
  end;
  suppose not V c= the_Vertices_of G2;
    then G1 == G2 by Def5;
    hence W is Walk of G2 by GLIB_001:179;
  end;
end;
