
theorem Th69: :: VS01
  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 holds S is VertexSeparator of b
  ,a
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;
A3: not b in S by A1,A2,Def8;
A4: for G2 being removeVertices of G,S holds not ex W being Walk of G2 st W
  is_Walk_from b,a
  proof
    let G2 be removeVertices of G,S;
    let W be Walk of G2;
    assume W is_Walk_from b,a;
    then W.reverse() is_Walk_from a,b by GLIB_001:23;
    hence contradiction by A1,A2,Def8;
  end;
  not a in S by A1,A2,Def8;
  hence thesis by A1,A2,A3,A4,Def8;
end;
