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 & not [y,x] in R by XBOOLE_0:def 5;
  then [x,y] in (P \ R)~ iff [x,y] in P~ & not [x,y] in R~ by Def5;
  hence thesis by XBOOLE_0:def 5;
end;
