
theorem Th72: :: VS08a
  for G being _Graph, a,b being Vertex of G st a<>b & not a,b
are_adjacent for S being VertexSeparator of a,b st S = {} holds not ex W being
  Walk of G st W is_Walk_from a,b
proof
  let G be _Graph;
  let a,b be Vertex of G such that
A1: a<>b and
A2: not a,b are_adjacent;
  let S be VertexSeparator of a,b;
  assume
A3: S = {};
A4: the_Edges_of G c= G.edgesBetween(the_Vertices_of G) by GLIB_000:34;
A5: the_Vertices_of G c= the_Vertices_of G;
  set G2 = the removeVertices of G,S;
  given W be Walk of G such that
A6: W is_Walk_from a,b;
  the_Vertices_of G2 = the_Vertices_of G by A3,GLIB_000:def 37;
  then
A7: W.vertices() c= the_Vertices_of G2;
  G2 is inducedSubgraph of G,the_Vertices_of G,the_Edges_of G by A3,GLIB_000:34
;
  then the_Edges_of G2 = the_Edges_of G by A5,A4,GLIB_000:def 37;
  then W.edges() c= the_Edges_of G2;
  then reconsider W2=W as Walk of G2 by A7,GLIB_001:170;
  W.last() = b by A6;
  then
A8: W2.last()=b;
  W.first() = a by A6;
  then W2.first()=a;
  then W2 is_Walk_from a,b by A8;
  hence contradiction by A1,A2,Def8;
end;
