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 (ln*arccos) & Z c= ]. -1,1 .[ & (for x st x in Z holds arccos
.x>0) implies ln*(arccos) is_differentiable_on Z & for x st x in Z holds ((ln*(
  arccos))`|Z).x=-1 / (sqrt(1-x^2)*arccos.x)
proof
  assume that
A1: Z c= dom (ln*(arccos)) and
A2: Z c= ]. -1,1 .[ and
A3: for x st x in Z holds arccos.x > 0;
A4: for x st x in Z holds ln*(arccos) is_differentiable_in x
  proof
    let x;
    assume x in Z;
    then arccos is_differentiable_in x & arccos.x >0 by A2,A3,FDIFF_1:9
,SIN_COS6:106;
    hence thesis by TAYLOR_1:20;
  end;
  then
A5: ln*arccos is_differentiable_on Z by A1,FDIFF_1:9;
  for x st x in Z holds ((ln*arccos)`|Z).x = -1 / (sqrt(1-x^2)*arccos.x)
  proof
    let x;
    assume
A6: x in Z;
    then
A7: -1 < x & x < 1 by A2,XXREAL_1:4;
    arccos is_differentiable_in x & arccos.x >0 by A2,A3,A6,FDIFF_1:9
,SIN_COS6:106;
    then diff(ln*arccos,x) =diff(arccos,x)/arccos.x by TAYLOR_1:20
      .=(-1 / sqrt(1-x^2))/arccos.x by A7,SIN_COS6:106
      .=-(1 / sqrt(1-x^2))/arccos.x by XCMPLX_1:187
      .=-1 / (sqrt(1-x^2)*arccos.x) by XCMPLX_1:78;
    hence thesis by A5,A6,FDIFF_1:def 7;
  end;
  hence thesis by A1,A4,FDIFF_1:9;
end;
