
theorem Th74: :: VS11
  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 being Vertex of G2 holds G2.reachableFrom(a2) /\ 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, G2 be removeVertices of G,S;
  let a2 be Vertex of G2;
  set A = G2.reachableFrom(a2);
  not a in S by A1,A2,Def8;
  then a in the_Vertices_of G \ S by XBOOLE_0:def 5;
  then
A3: the_Vertices_of G2 = the_Vertices_of G \ S by GLIB_000:def 37;
  now
    let x be object such that
A4: x in A /\ S;
A5: x in S by A4,XBOOLE_0:def 4;
    x in A by A4,XBOOLE_0:def 4;
    hence contradiction by A3,A5,XBOOLE_0:def 5;
  end;
  hence thesis by XBOOLE_0:def 1;
end;
