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=]. -1,1 .[ & Z c= dom (r(#)arccos) implies r(#)arccos
is_differentiable_on Z & for x st x in Z holds ((r(#)arccos)`|Z).x =- r / sqrt(
  1-x^2)
proof
  assume that
A1: Z c= ]. -1,1 .[ and
A2: Z c= dom (r(#)arccos);
A3: arccos is_differentiable_on Z by A1,FDIFF_1:26,SIN_COS6:106;
  for x st x in Z holds ((r(#)arccos)`|Z).x = -r / sqrt(1-x^2)
  proof
    let x;
    assume
A4: x in Z;
    then
A5: -1 < x & x < 1 by A1,XXREAL_1:4;
    ((r(#)arccos)`|Z).x = r*diff(arccos,x) by A2,A3,A4,FDIFF_1:20
      .= r*(-1 / sqrt(1-x^2)) by A5,SIN_COS6:106
      .=-r*(1 / sqrt(1-x^2))
      .=-r / sqrt(1-x^2) by XCMPLX_1:99;
    hence thesis;
  end;
  hence thesis by A2,A3,FDIFF_1:20;
end;
