
theorem
  for P, Q being pcs holds P misses Q implies pcs-sum(P,Q) is pcs
proof
  let P, Q be pcs;
  assume
A1: P misses Q;
  set R = pcs-sum(P,Q);
A2: field the InternalRel of R = the carrier of R by ORDERS_1:12;
  the InternalRel of R is transitive by A1,Th18;
  then the InternalRel of R is_transitive_in the carrier of R
  by A2;
  then
A3: R is transitive;
  R is pcs-compatible by A1,Th19;
  hence thesis by A3;
end;
