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
  R|(X /\ Y) = (R|X) /\ (R|Y)
  proof
    let x,y;
    hereby
      assume
A1:   [x,y] in R|(X /\ Y); then
A2:   x in X /\ Y by Def9;
A3:   [x,y] in R by A1,Def9;
      x in Y by A2,XBOOLE_0:def 4; then
A4:   [x,y] in R|Y by A3,Def9;
      x in X by A2,XBOOLE_0:def 4;
      then [x,y] in R|X by A3,Def9;
      hence [x,y] in (R|X) /\ (R|Y) by A4,XBOOLE_0:def 4;
    end;
    assume
A5: [x,y] in (R|X) /\ (R|Y);
    then [x,y] in R|Y by XBOOLE_0:def 4; then
A6: x in Y by Def9;
A7: [x,y] in R|X by A5,XBOOLE_0:def 4;
    then x in X by Def9; then
A8: x in X /\ Y by A6,XBOOLE_0:def 4;
    [x,y] in R by A7,Def9;
    hence [x,y] in R|(X /\ Y) by A8,Def9;
  end;
