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*Q) c= P*(R \ Q)
proof
  let a,b;
  assume
A1: [a,b] in (P*R) \ (P*Q);
  then consider y such that
A2: [a,y] in P and
A3: [y,b] in R by Def6;
  not [a,b] in P*Q by A1,XBOOLE_0:def 5;
  then not [a,y] in P or not [y,b] in Q by Def6;
  then [y,b] in R \ Q by A2,A3,XBOOLE_0:def 5;
  hence [a,b] in P*(R \ Q) by A2,Def6;
end;
