
theorem Th4:
  for f,g being Function, x,X being set holds
  x in X & X c= dom f implies ((f,X)+*g).x = f.x
proof
  let f,g be Function;
  let x,X be set;
  assume
A1: x in X;
  assume
A2: X c= dom f;
  dom (f|X) = dom f /\ X by RELAT_1:61;
  then
A3: x in dom (f|X) by A1,A2,XBOOLE_0:def 4;
  then (f|X).x = f.x by FUNCT_1:47;
  hence thesis by A3,FUNCT_4:13;
end;
