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 Th1:
  cos.x<>0 implies sec is_differentiable_in x & diff(sec, x) = sin. x/(cos.x)^2
proof
A1: cos is_differentiable_in x by SIN_COS:63;
  assume
A2: cos.x<>0;
  then diff(cos^,x)= - diff(cos,x)/(cos.x)^2 by A1,FDIFF_2:15
    .=- (-sin.x)/(cos.x)^2 by SIN_COS:63
    .=sin.x/(cos.x)^2;
  hence thesis by A2,A1,FDIFF_2:15;
end;
