reserve x for Real,

  n for Element of NAT,
   y for set,
  Z for open Subset of REAL,

     g for PartFunc of REAL,REAL;

theorem
  Z c= dom sec & Z c= ].-1,1.[ implies sec(#)(arctan-arccot)
is_differentiable_on Z & for x st x in Z holds ((sec(#)(arctan-arccot))`|Z).x =
  (arctan.x-arccot.x)*sin.x/(cos.x)^2+2*sec.x/(1+x^2)
proof
  assume that
A1: Z c= dom sec and
A2: Z c= ].-1,1.[;
A3: arctan-arccot is_differentiable_on Z by A2,Th38;
  for x st x in Z holds sec is_differentiable_in x
  proof
    let x;
    assume x in Z;
    then cos.x <> 0 by A1,RFUNCT_1:3;
    hence thesis by FDIFF_9:1;
  end;
  then
A4: sec is_differentiable_on Z by A1,FDIFF_1:9;
A5: ].-1,1.[ c= [.-1,1.] by XXREAL_1:25;
  then ].-1,1.[ c= dom arccot by SIN_COS9:24,XBOOLE_1:1;
  then
A6: Z c= dom arccot by A2,XBOOLE_1:1;
  ].-1,1.[ c= dom arctan by A5,SIN_COS9:23,XBOOLE_1:1;
  then Z c= dom arctan by A2,XBOOLE_1:1;
  then Z c= dom arctan /\ dom arccot by A6,XBOOLE_1:19;
  then
A7: Z c= dom (arctan-arccot) by VALUED_1:12;
  then Z c= dom sec /\ dom (arctan-arccot) by A1,XBOOLE_1:19;
  then
A8: Z c= dom (sec(#)(arctan-arccot)) by VALUED_1:def 4;
  for x st x in Z holds ((sec(#)(arctan-arccot))`|Z).x = (arctan.x-arccot
  .x)*sin.x/(cos.x)^2+2*sec.x/(1+x^2)
  proof
    let x;
    assume
A9: x in Z;
    then
A10: cos.x <> 0 by A1,RFUNCT_1:3;
    (sec(#)(arctan-arccot)`|Z).x = ((arctan-arccot).x)*diff(sec,x)+sec.x*
    diff(arctan-arccot,x) by A8,A4,A3,A9,FDIFF_1:21
      .= (arctan.x-arccot.x)*diff(sec,x)+sec.x*diff(arctan-arccot,x) by A7,A9,
VALUED_1:13
      .= (arctan.x-arccot.x)*(sin.x/(cos.x)^2)+sec.x*diff(arctan-arccot,x)
    by A10,FDIFF_9:1
      .= (arctan.x-arccot.x)*sin.x/(cos.x)^2+sec.x*((arctan-arccot)`|Z).x by A3
,A9,FDIFF_1:def 7
      .= (arctan.x-arccot.x)*sin.x/(cos.x)^2+sec.x*(2/(1+x^2)) by A2,A9,Th38
      .= (arctan.x-arccot.x)*sin.x/(cos.x)^2+2*sec.x/(1+x^2);
    hence thesis;
  end;
  hence thesis by A8,A4,A3,FDIFF_1:21;
end;
