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 /\ Q) c= (P*R) /\ (P*Q)
proof
  let x,y;
  assume [x,y] in P*(R /\ Q);
  then consider z such that
A1: [x,z] in P and
A2: [z,y] in R /\ Q by Def6;
  [z,y] in Q by A2,XBOOLE_0:def 4;
  then
A3: [x,y] in P*Q by A1,Def6;
  [z,y] in R by A2,XBOOLE_0:def 4;
  then [x,y] in P*R by A1,Def6;
  hence [x,y] in (P*R) /\ (P*Q) by A3,XBOOLE_0:def 4;
end;
