reserve x,r,a,x0,p for Real;
reserve n,i,m for Element of NAT;
reserve Z for open Subset of REAL;
reserve f,f1,f2 for PartFunc of REAL,REAL;
reserve k for Nat;

theorem Th1:
  for f being Function of REAL, REAL holds dom (f | Z) = Z
proof
  let f be Function of REAL, REAL;
A1: dom f = REAL by FUNCT_2:def 1;
  thus dom (f|Z) = dom f /\ Z by RELAT_1:61
    .= Z by A1,XBOOLE_1:28;
end;
