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= ].-1,1.[ implies sin(#)(arctan+arccot) is_differentiable_on Z &
for x st x in Z holds ((sin(#)(arctan+arccot))`|Z).x = cos.x*(arctan.x+arccot.x
  )
proof
  for x st x in Z holds sin is_differentiable_in x by SIN_COS:64;
  then
A1: sin is_differentiable_on Z by FDIFF_1:9,SIN_COS:24;
  assume
A2: Z c= ].-1,1.[;
  then
A3: arctan+arccot is_differentiable_on Z by Th37;
A4: ].-1,1.[ c= [.-1,1.] by XXREAL_1:25;
  then ].-1,1.[ c= dom arccot by SIN_COS9:24,XBOOLE_1:1;
  then
A5: Z c= dom arccot by A2,XBOOLE_1:1;
  ].-1,1.[ c= dom arctan by A4,SIN_COS9:23,XBOOLE_1:1;
  then Z c= dom arctan by A2,XBOOLE_1:1;
  then Z c= dom arctan /\ dom arccot by A5,XBOOLE_1:19;
  then
A6: Z c= dom (arctan+arccot) by VALUED_1:def 1;
  then Z c= dom sin /\ dom (arctan+arccot) by SIN_COS:24,XBOOLE_1:19;
  then
A7: Z c= dom (sin(#)(arctan+arccot)) by VALUED_1:def 4;
  for x st x in Z holds (sin(#)(arctan+arccot)`|Z).x = cos.x*(arctan.x+
  arccot.x)
  proof
    let x;
    assume
A8: x in Z;
    then
    (sin(#)(arctan+arccot)`|Z).x = ((arctan+arccot).x)*diff(sin,x)+sin.x*
    diff(arctan+arccot,x) by A7,A1,A3,FDIFF_1:21
      .= (arctan.x+arccot.x)*diff(sin,x)+sin.x*diff(arctan+arccot,x) by A6,A8,
VALUED_1:def 1
      .= (arctan.x+arccot.x)*cos.x+sin.x*diff(arctan+arccot,x) by SIN_COS:64
      .= (arctan.x+arccot.x)*cos.x+sin.x*((arctan+arccot)`|Z).x by A3,A8,
FDIFF_1:def 7
      .= (arctan.x+arccot.x)*cos.x+sin.x*0 by A2,A8,Th37
      .= cos.x*(arctan.x+arccot.x);
    hence thesis;
  end;
  hence thesis by A7,A1,A3,FDIFF_1:21;
end;
