
theorem
  for G being _Graph, a,b being Vertex of G st a<>b &not a,b
  are_adjacent & not ex W being Walk of G st W is_Walk_from a,b holds {} is
  VertexSeparator of 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;
  assume
A3: not ex W being Walk of G st W is_Walk_from a,b;
A4: now
    let G2 be removeVertices of G,{};
    given W be Walk of G2 such that
A5: W is_Walk_from a,b or W is_Walk_from b,a;
    per cases by A5;
    suppose
A6:   W is_Walk_from a,b;
      reconsider W2=W as Walk of G by GLIB_001:167;
      W.last() = b by A6;
      then
A7:   W2.last()=b;
      W.first() = a by A6;
      then W2.first()=a;
      then W2 is_Walk_from a,b by A7;
      hence contradiction by A3;
    end;
    suppose
A8:   W is_Walk_from b,a;
      set P=W.reverse();
      reconsider W2=P as Walk of G by GLIB_001:167;
A9:   P is_Walk_from a,b by A8,GLIB_001:23;
      then P.last() = b;
      then
A10:  W2.last()=b;
      P.first() = a by A9;
      then W2.first()=a;
      then W2 is_Walk_from a,b by A10;
      hence contradiction by A3;
    end;
  end;
  {} is Subset of the_Vertices_of G by XBOOLE_1:2;
  hence thesis by A1,A2,A4,Def8;
end;
