reserve B for non empty set,
  A,X,x for set;
reserve phi for Element of Funcs(A,B);

theorem Th5:
  phi|X = phi|(A /\ X)
proof
  dom phi = A by FUNCT_2:def 1;
  then
A1: dom (phi|X) = (dom phi /\ A) /\ X by RELAT_1:61
    .= dom phi /\ (A /\ X) by XBOOLE_1:16;
  for x being object st x in dom (phi|X) holds (phi|X).x = phi.x by
   FUNCT_1:47;
  hence thesis by A1,FUNCT_1:46;
end;
