
theorem
  for G being _finite chordal _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 H
  being removeVertices of G,S, a1 being Vertex of H st a = a1 for x, y being
  Vertex of G st x in S & y in S holds ex c being Vertex of G st c in H
  .reachableFrom(a1) & c,x are_adjacent & c,y are_adjacent
proof
  let G be _finite chordal _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 H be removeVertices of G,S, a1 be Vertex of H;
  assume a = a1;
  then consider c being Vertex of G such that
A4: c in H.reachableFrom(a1) and
A5: for x being Vertex of G st x in S holds c,x are_adjacent by A1,A2,A3,Th99;
  let x,y be Vertex of G such that
A6: x in S and
A7: y in S;
A8: c,y are_adjacent by A7,A5;
  c,x are_adjacent by A6,A5;
  hence thesis by A4,A8;
end;
