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) c= R.:X /\ R.:Y
proof
  let y be object;
  assume y in R.:(X /\ Y);
  then consider x such that
A1: [x,y] in R and
A2: x in X /\ Y by Def11;
  x in Y by A2,XBOOLE_0:def 4;
  then
A3: y in R.:Y by A1,Def11;
  x in X by A2,XBOOLE_0:def 4;
  then y in R.:X by A1,Def11;
  hence thesis by A3,XBOOLE_0:def 4;
end;
