reserve P,Q,X,Y,Z for set, p,x,x9,x1,x2,y,z for object;

theorem
  for f being Function of X,Y st Y <> {} & x in X & f.x in Z holds (Z|`f)
  .x = f.x
proof
  let f be Function of X,Y;
  assume that
A1: Y <> {} & x in X and
A2: f.x in Z;
  x in dom f by A1,Def1;
  then x in dom(Z|`f) by A2,FUNCT_1:54;
  hence thesis by FUNCT_1:55;
end;
