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
  field(P /\ R) c= field P /\ field R
proof
  let x be object;
  assume x in field(P /\ R);
  then
A1: x in dom(P /\ R) or x in rng(P /\ R) by XBOOLE_0:def 3;
  x in dom P /\ dom R or x in rng P /\ rng R implies (x in dom P or x in
  rng P) & (x in dom R or x in rng R) by XBOOLE_0:def 4;
  then
A2: x in dom P /\ dom R or x in rng P /\ rng R implies x in dom P \/ rng P &
  x in dom R \/ rng R by XBOOLE_0:def 3;
  dom(P /\ R) c= dom P /\ dom R & rng(P /\ R) c= rng P /\ rng R
  by XTUPLE_0:24,28;
  hence thesis by A1,A2,XBOOLE_0:def 4;
end;
