reserve x,a,b,c for Real,
  n for Nat,
  Z for open Subset of REAL,
  f, f1,f2 for PartFunc of REAL,REAL;

theorem
  Z c= dom sec implies sec is_differentiable_on Z & for x st x in Z
  holds ( (sec)`|Z).x = sin.x/(cos.x)^2
proof
  assume Z c= dom sec;
  then for x st x in Z holds cos.x<>0 by RFUNCT_1:3;
  hence thesis by FDIFF_4:39;
end;
