theorem
  (P /\ R)~ = P~ /\ R~
proof
  let x,y;
  [x,y] in (P /\ R)~ iff [y,x] in P /\ R by Def5;
  then [x,y] in (P /\ R)~ iff [y,x] in P & [y,x] in R by XBOOLE_0:def 4;
  then [x,y] in (P /\ R)~ iff [x,y] in P~ & [x,y] in R~ by Def5;
  hence thesis by XBOOLE_0:def 4;
end;
