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 Th17:
  Z c= dom ((id Z)(#)(arccos)) & Z c= ]. -1,1 .[ implies (id Z)(#)
(arccos) is_differentiable_on Z & for x st x in Z holds (((id Z)(#)(arccos))`|Z
  ).x =arccos.x-x/sqrt(1-x^2)
proof
  assume that
A1: Z c= dom ((id Z)(#)(arccos)) and
A2: Z c= ]. -1,1 .[;
A3: for x st x in Z holds (id Z).x = 1*x+0 by FUNCT_1:18;
  Z c= dom (id Z) /\ dom (arccos) by A1,VALUED_1:def 4;
  then
A4: Z c= dom (id Z) by XBOOLE_1:18;
  then
A5: id Z is_differentiable_on Z by A3,FDIFF_1:23;
A6: arccos is_differentiable_on Z by A2,FDIFF_1:26,SIN_COS6:106;
  for x st x in Z holds (((id Z)(#)(arccos))`|Z).x =arccos.x-x/sqrt(1-x^2)
  proof
    let x;
    assume
A7: x in Z;
    then
A8: -1 < x & x < 1 by A2,XXREAL_1:4;
    (((id Z)(#)(arccos))`|Z).x =(arccos.x)*diff((id Z),x) + ((id Z).x)*
    diff(arccos,x) by A1,A5,A6,A7,FDIFF_1:21
      .=(arccos.x)*((id Z)`|Z).x+ ((id Z).x)*diff(arccos,x) by A5,A7,
FDIFF_1:def 7
      .=(arccos.x)*1+ ((id Z).x)*diff(arccos,x) by A4,A3,A7,FDIFF_1:23
      .=(arccos.x)*1+ ((id Z).x)*(-1 / sqrt(1-x^2)) by A8,SIN_COS6:106
      .=arccos.x+ x*(-1 / sqrt(1-x^2)) by A7,FUNCT_1:18
      .=arccos.x-x*(1 / sqrt(1-x^2))
      .=arccos.x-x / sqrt(1-x^2) by XCMPLX_1:99;
    hence thesis;
  end;
  hence thesis by A1,A5,A6,FDIFF_1:21;
end;
