
theorem Th23: :: Walk04
  for G being _Graph,
      S being non empty Subset of the_Vertices_of G
  for H being inducedSubgraph of G,S for W being Walk of G st
    W.vertices() c= S holds W is Walk of H
proof
  let G be _Graph, S be non empty Subset of the_Vertices_of G;
  let H be inducedSubgraph of G,S;
A1: the_Vertices_of H = S by GLIB_000:def 37;
A2: the_Edges_of H = G.edgesBetween(S) by GLIB_000:def 37;
  let W be Walk of G;
  assume W.vertices() c= S;
  then
A3: W.vertices() c= the_Vertices_of H by GLIB_000:def 37;
A4: W.edges() c= G.edgesBetween(W.vertices()) by GLIB_001:109;
  G.edgesBetween(W.vertices()) c= G.edgesBetween(the_Vertices_of H) by A3,
GLIB_000:36;
  then W.edges() c= the_Edges_of H by A1,A2,A4;
  hence thesis by A3,GLIB_001:170;
end;
