
theorem Th31:
  for P being pcs-compatible pcs-Str, a being set st not a in the carrier of P
  holds pcs-extension(P,a) is pcs-compatible
proof
  let P be pcs-compatible pcs-Str, a be set such that
A1: not a in the carrier of P;
  set R = pcs-extension(P,a);
  let p, p9, q, q9 be Element of R such that
A2: p (--) q and
A3: p9 <= p and
A4: q9 <= q;
  per cases;
  suppose p9 = a or q9 = a;
    hence thesis by Th27;
  end;
  suppose that
A5: p9 <> a and
A6: q9 <> a;
    reconsider p90 = p9, q90 = q9 as Element of P by A5,A6,Th25;
A7: p90 <> a by A5;
A8: q90 <> a by A6;
A9: p <> a by A1,A3,A7,Th24;
A10: q <> a by A1,A4,A8,Th24;
    reconsider p0 = p, q0 = q as Element of P by A1,A3,A4,A7,A8,Th24;
A11: p0 (--) q0 by A2,A9,A10,Th29;
A12: p90 <= p0 by A3,A5,Th26;
    q90 <= q0 by A4,A6,Th26;
    then p90 (--) q90 by A11,A12,Def22;
    hence thesis by Th28;
  end;
end;
