
theorem :: VS06
  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 is minimal for x being
  Vertex of G st x in S ex W being Walk of G st W is_Walk_from a,b & x in W
  .vertices()
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 such that
A3: S is minimal;
  let x be Vertex of G such that
A4: x in S;
  set T = S\{x};
A5: T c= S by XBOOLE_1:36;
  then
A6: not b in T by A1,A2,Def8;
  assume
A7: not ex W being Walk of G st W is_Walk_from a,b & x in W.vertices();
A8: now
    let W be Walk of G such that
A9: W is_Walk_from a,b;
    consider y being Vertex of G such that
A10: y in S and
A11: y in W.vertices() by A1,A2,A9,Th70;
    take y;
    y <> x by A7,A9,A11;
    then not y in {x} by TARSKI:def 1;
    hence y in T by A10,XBOOLE_0:def 5;
    thus y in W.vertices() by A11;
  end;
  x in {x} by TARSKI:def 1;
  then
A12: T <> S by A4,XBOOLE_0:def 5;
  not a in T by A1,A2,A5,Def8;
  then T is VertexSeparator of a,b by A1,A2,A6,A8,Th70;
  hence contradiction by A3,A12,A5;
end;
