reserve A,X,X1,X2,Y,Y1,Y2 for set, a,b,c,d,x,y,z for object;
reserve P,P1,P2,Q,R,S for Relation;

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 or [y,x] in R by XBOOLE_0:def 3;
  then [x,y] in (P \/ R)~ iff [x,y] in P~ or [x,y] in R~ by Def5;
  hence thesis by XBOOLE_0:def 3;
end;
