theorem
  p ||= PP_or(q,r) implies
  for d st d in dom p & p.d = TRUE holds
   d in dom q & q.d = TRUE or d in dom r & r.d = TRUE
  proof
    assume that
A1: p ||= PP_or(q,r);
    let d;
    assume d in dom p & p.d = TRUE;
    then d in dom PP_or(q,r) & PP_or(q,r).d = TRUE by A1;
    hence thesis by PARTPR_1:10;
  end;
