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
  (X \ Y)|`R = X|`R \ Y|`R
proof
  let x,y;
A1: y in X \ Y iff y in X & not y in Y by XBOOLE_0:def 5;
A2: [x,y] in X|`R \ Y|`R iff [x,y] in X|`R & not [x,y] in Y|`R
          by XBOOLE_0:def 5;
  [x,y] in X|`R iff y in X & [x,y] in R by Def10;
  hence thesis by A1,A2,Def10;
end;
