
theorem
  for P being pcs, a being set st not a in the carrier of P
  holds pcs-extension(P,a) is pcs
proof
  let P be pcs, a be set such that
A1: not a in the carrier of P;
  set R = pcs-extension(P,a);
  R is reflexive transitive pcs-tol-reflexive pcs-tol-symmetric
  pcs-compatible by A1,Th30,Th31;
  hence thesis;
end;
