reserve A for set,
  C for non empty set,
  B for Subset of A,
  x for Element of A,
  f,g for Function of A,C;

theorem Th1:
  dom (g|B) = B
proof
  thus dom(g|B) = dom g /\ B by RELAT_1:61
    .= A /\ B by FUNCT_2:def 1
    .= B by XBOOLE_1:28;
end;
