reserve x for object;
reserve D for set;
reserve p for PartialPredicate of D;
reserve D for non empty set;
reserve p,q,r for PartialPredicate of D;

theorem
  PP_or(p,PP_and(q,r)) = PP_and(PP_or(p,q),PP_or(p,r))
  proof
    reconsider a = p, b = q, c = r
     as Element of PartialPredConnectivesLatt(D) by PARTFUN1:45;
A1: PP_or(p,q) = a"\/"b & PP_or(p,r) = a"\/"c by Def12;
    PP_and(q,r) = b"/\"c by Def11;
    then
A2: PP_or(p,PP_and(q,r)) = a"\/"(b"/\"c) by Def12;
    PP_and(PP_or(p,q),PP_or(p,r)) = (a"\/"b)"/\"(a"\/"c) by A1,Def11;
    hence thesis by A2,LATTICES:11;
  end;
