reserve v,x for object;
reserve D,V,A for set;
reserve n for Nat;
reserve p,q for PartialPredicate of D;
reserve f,g for BinominativeFunction of D;
reserve D for non empty set;
reserve d for Element of D;
reserve f,g for BinominativeFunction of D;
reserve p,q,r,s for PartialPredicate of D;

theorem
  p ||= q & r ||= s implies PP_and(p,r) ||= PP_and(q,s)
  proof
    assume that
A1: p ||= q and
A2: r ||= s;
    let d such that
A3: d in dom PP_and(p,r) & PP_and(p,r).d = TRUE;
A4: dom(PP_and(q,s)) =
    {d where d is Element of D:
    d in dom q & q.d = FALSE or d in dom s & s.d = FALSE
    or d in dom q & q.d = TRUE & d in dom s & s.d = TRUE} by PARTPR_1:16;
    d in dom p & p.d = TRUE & d in dom r & r.d = TRUE by A3,PARTPR_1:23;
    then d in dom q & q.d = TRUE & d in dom s & s.d = TRUE by A1,A2;
    hence thesis by A4,PARTPR_1:18;
  end;
