reserve P,Q,X,Y,Z for set, p,x,x9,x1,x2,y,z for object;
reserve D for non empty set;
reserve A,B for non empty set;

theorem
  for f being Function, A0, D being set st f"D c= A0 holds f"D = (f|A0)" D
proof
  let f be Function, A0, D be set;
  assume
A1: f"D c= A0;
  thus (f|A0)"D = (f*(id A0))"D by RELAT_1:65
    .= (id A0)"(f"D) by RELAT_1:146
    .= f"D by A1,Th93;
end;
