reserve a,b,c,d,x,y,z for object, X,Y,Z for set;
reserve R,S,T for Relation;
reserve F,G for Function;

theorem Th12:
  x in field(R |_2 X) implies x in field R & x in X
proof
A1: dom(X|`R|X) = dom(X|`R) /\ X & rng(X|`(R|X)) = rng(R|X) /\ X
             by RELAT_1:61,88;
  assume x in field(R |_2 X);
  then
A2: x in dom(R |_2 X) or x in rng(R |_2 X) by XBOOLE_0:def 3;
A3: dom(X|`R) c= dom R & rng(R|X) c= rng R by Lm5,RELAT_1:70;
  R |_2 X = X|`R|X & R |_2 X = X|`(R|X) by Th10,Th11;
  then x in dom(X|`R) & x in X or x in rng(R|X) & x in X
    by A2,A1,XBOOLE_0:def 4;
  hence thesis by A3,XBOOLE_0:def 3;
end;
