reserve f,f1,f2,g for PartFunc of REAL,REAL;
reserve A for non empty closed_interval Subset of REAL;
reserve p,r,x,x0 for Real;
reserve n for Element of NAT;
reserve Z for open Subset of REAL;

theorem Th33:
  Z c= dom tan implies tan is_differentiable_on Z & for x st x in Z holds
  ((tan)`|Z).x = 1/(cos.x)^2
proof
  assume that
A1: Z c= dom tan;
A2: for x st x in Z holds tan is_differentiable_in x
  proof
    let x;
    assume x in Z; then
    x in dom tan by A1; then
A3: cos.x <> 0 by FDIFF_8:1;
    sin is_differentiable_in x & cos is_differentiable_in x by SIN_COS:63,64;
    hence thesis by A3,FDIFF_2:14;
  end; then
A4: tan is_differentiable_on Z by A1,FDIFF_1:9;
  for x st x in Z holds ((tan)`|Z).x = 1/(cos.x)^2
  proof
    let x;
A5: sin is_differentiable_in x & cos is_differentiable_in x by SIN_COS:63,64;
    assume
A6: x in Z; then
    x in dom tan by A1;
    then cos.x <> 0 by FDIFF_8:1;
    then
    diff (tan,x) = (diff(sin,x) * cos.x - diff(cos,x)*sin.x)/(cos.x)^2 by A5,
FDIFF_2:14
      .=((cos.x)*cos.x - diff(cos,x)*sin.x)/(cos.x)^2 by SIN_COS:64
      .=((cos.x)*(cos.x)-(-sin.x)*(sin.x))/(cos.x)^2 by SIN_COS:63
      .=((cos.x)*(cos.x) + (sin.x)*(sin.x))/(cos.x)^2
      .=1/(cos.x)^2 by SIN_COS:28;
    hence thesis by A4,A6,FDIFF_1:def 7;
  end;
  hence thesis by A1,A2,FDIFF_1:9;
end;
