
theorem Th71: :: VS07
  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 for W being Walk of G st W
  is_Walk_from a,b ex k being odd Nat st 1 < k & k < len W & W.k in S
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;
  let W be Walk of G such that
A3: W is_Walk_from a,b;
  consider x being Vertex of G such that
A4: x in S and
A5: x in W.vertices() by A1,A2,A3,Th70;
  consider n being odd Element of NAT such that
A6: n <= len W and
A7: W.n = x by A5,GLIB_001:87;
    not a in S by A1,A2,Th70;
    then
A8: 1 <> n by A3,A4,A7;
    not b in S by A1,A2,Th70;
   then n <> len W by A3,A4,A7;
  then
A9: n < len W by A6,XXREAL_0:1;
  1 <= n by ABIAN:12;
  then 1 < n by A8,XXREAL_0:1;
  hence thesis by A4,A7,A9;
end;
