reserve y for set,
  x,r,a,b for Real,
  n for Element of NAT,
  Z for open Subset of REAL,
  f,f1,f2,f3 for PartFunc of REAL,REAL;

theorem
  Z c= dom ((1/2)(#)(( #Z 2)*(arccos))) & Z c=]. -1,1 .[ implies (1/2)
(#)(( #Z 2)*(arccos)) is_differentiable_on Z & for x st x in Z holds (((1/2)(#)
  (( #Z 2)*(arccos)))`|Z).x =-arccos.x / sqrt(1-x^2)
proof
  assume that
A1: Z c= dom ((1/2)(#)(( #Z 2)*(arccos))) and
A2: Z c=]. -1,1 .[;
A3: Z c= dom (( #Z 2)*(arccos)) by A1,VALUED_1:def 5;
  then
A4: ( #Z 2)*(arccos) is_differentiable_on Z by A2,Th11;
  for x st x in Z holds (((1/2)(#)(( #Z 2)*(arccos)))`|Z).x =-arccos.x /
  sqrt(1-x^2)
  proof
    let x;
    assume
A5: x in Z;
    then
    (((1/2)(#)(( #Z 2)*(arccos)))`|Z).x =(1/2)*diff((( #Z 2)*arccos),x) by A1
,A4,FDIFF_1:20
      .=(1/2)*((( #Z 2)*(arccos))`|Z).x by A4,A5,FDIFF_1:def 7
      .=(1/2)*(-2*(arccos.x) #Z (2-1) / sqrt(1-x^2)) by A2,A3,A5,Th11
      .=-(1/2)*((2*(arccos.x) #Z (2-1)) / sqrt(1-x^2))
      .=-((1/2)*(2*(arccos.x) #Z (2-1))) / sqrt(1-x^2) by XCMPLX_1:74
      .=-arccos.x / sqrt(1-x^2) by PREPOWER:35;
    hence thesis;
  end;
  hence thesis by A1,A4,FDIFF_1:20;
end;
