
theorem Th75:
  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, G2 being removeVertices of G,S
  for a2,b2 being Vertex of G2 st a2=a & b2=b holds G2.reachableFrom(a2) /\ G2
  .reachableFrom(b2) = {}
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, G2 be removeVertices of G,S;
  let a2,b2 be Vertex of G2 such that
A3: a2=a and
A4: b2=b;
  set A = G2.reachableFrom(a2), B = G2.reachableFrom(b2);
  now
    let x be object such that
A5: x in A /\ B;
    x in A by A5,XBOOLE_0:def 4;
    then consider W1 being Walk of G2 such that
A6: W1 is_Walk_from a2,x by GLIB_002:def 5;
    x in B by A5,XBOOLE_0:def 4;
    then consider rW2 being Walk of G2 such that
A7: rW2 is_Walk_from b2,x by GLIB_002:def 5;
    set W2 = rW2.reverse();
    set W = W1.append(W2);
    W2 is_Walk_from x,b2 by A7,GLIB_001:23;
    then W is_Walk_from a2,b2 by A6,GLIB_001:31;
    hence contradiction by A1,A2,A3,A4,Def8;
  end;
  hence thesis by XBOOLE_0:def 1;
end;
