
theorem Th21: :: Walk02
  for G being _Graph, S being Subset of the_Vertices_of G
  for H being removeVertices of G,S for W being Walk of G st
  (for n being odd Nat st n <= len W holds not W.n in S)
  holds W is Walk of H
proof
  let G be _Graph, S be Subset of the_Vertices_of G;
  let H be removeVertices of G,S;
  let W be Walk of G such that
A1: for n being odd Nat st n <= len W holds not W.n in S;
A2: now
    assume the_Vertices_of G\S = {};
    then
A3: the_Vertices_of G c= S by XBOOLE_1:37;
    W.last() in the_Vertices_of G;
    hence contradiction by A1,A3;
  end;
  then
A4: the_Edges_of H = G.edgesBetween(the_Vertices_of G\S) by GLIB_000:def 37;
A5: W.edges() c= G.edgesBetween(W.vertices()) by GLIB_001:109;
A6: the_Vertices_of H = the_Vertices_of G\S by A2,GLIB_000:def 37;
  now
    let x be object such that
A7: x in W.vertices();
    ex n being odd Element of NAT st n <= len W & W.n = x by A7,GLIB_001:87;
    then not x in S by A1;
    hence x in the_Vertices_of H by A6,A7,XBOOLE_0:def 5;
  end;
  then
A8: W.vertices() c= the_Vertices_of H;
  then G.edgesBetween(W.vertices()) c= G.edgesBetween(the_Vertices_of H) by
GLIB_000:36;
  then W.edges() c= G.edgesBetween(the_Vertices_of H) by A5;
  hence thesis by A6,A4,A8,GLIB_001:170;
end;
