
theorem
  for G being _finite _Graph holds ((LexBFS:CSeq(G)).Result())`1 is
  with_property_T
proof
  let G be _finite _Graph;
  set CS = LexBFS:CSeq(G);
  set L = (CS.Result())`1;
A1: L is with_property_L3 by Th53;
  now
    let a,b,c be Vertex of G such that
A2: a in dom L and
A3: b in dom L and
A4: c in dom L and
A5: L.a < L.b and
A6: L.b < L.c and
A7: a,c are_adjacent and
A8: not b,c are_adjacent;
    consider d being Vertex of G such that
A9: d in dom L and
A10: L.c < L.d and
A11: b,d are_adjacent and
A12: not a,d are_adjacent and
    for e being Vertex of G st e <> d & e,b are_adjacent & not e,a
    are_adjacent holds L.e < L.d by A1,A2,A3,A4,A5,A6,A7,A8;
    take d;
    thus d in dom L by A9;
    thus L.b < L.d by A6,A10,XXREAL_0:2;
    thus b,d are_adjacent by A11;
    thus not a,d are_adjacent by A12;
  end;
  hence thesis;
end;
