reserve n,m for Nat,
  r,r1,r2,s,t for Real,
  x,y for set;

theorem
  for D be non empty set, F be PartFunc of D,REAL, r be Real, X be set
  holds (F|X) - r = (F-r)| X
proof
  let D be non empty set, F be PartFunc of D,REAL, r be Real, X be set;
A1: dom ((F|X) - r) = dom (F|X) by VALUED_1:3;
A2: dom (F|X) = dom F /\ X by RELAT_1:61;
A3: dom F /\ X = dom(F-r) /\ X by VALUED_1:3
    .= dom((F-r)|X) by RELAT_1:61;
  now
    let d be Element of D;
    assume
A4: d in dom ((F|X) - r);
    then
A5: d in dom F by A1,A2,XBOOLE_0:def 4;
    thus ((F|X) - r).d = (F|X).d - r by A1,A4,VALUED_1:3
      .= F.d -r by A1,A4,FUNCT_1:47
      .= (F-r).d by A5,VALUED_1:3
      .=((F-r)|X).d by A1,A2,A3,A4,FUNCT_1:47;
  end;
  hence thesis by A2,A3,PARTFUN1:5,VALUED_1:3;
end;
