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 Th90:
  Y|`(X|`R) = (Y /\ X)|`R
proof
  let x,y;
A1: [x,y] in X|`R iff [x,y] in R & y in X by Def10;
A2: [x,y] in (Y /\ X)|`R iff [x,y] in R & y in Y /\ X by Def10;
  [x,y] in Y|`(X|`R) iff [x,y] in X|`R & y in Y by Def10;
  hence thesis by A1,A2,XBOOLE_0:def 4;
end;
