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*(arcsin)) & Z c= ]. -1,1 .[ & (for x st x in Z holds
arcsin.x>0) implies ln*(arcsin) is_differentiable_on Z & for x st x in Z holds
  ((ln*(arcsin))`|Z).x=1 / (sqrt(1-x^2)*arcsin.x)
proof
  assume that
A1: Z c= dom (ln*arcsin) and
A2: Z c= ]. -1,1 .[ and
A3: for x st x in Z holds arcsin.x > 0;
A4: for x st x in Z holds ln*arcsin is_differentiable_in x
  proof
    let x;
    assume x in Z;
    then arcsin is_differentiable_in x & arcsin.x >0 by A2,A3,FDIFF_1:9
,SIN_COS6:83;
    hence thesis by TAYLOR_1:20;
  end;
  then
A5: ln*arcsin is_differentiable_on Z by A1,FDIFF_1:9;
  for x st x in Z holds ((ln*(arcsin))`|Z).x = 1 / (sqrt(1-x^2)*arcsin.x)
  proof
    let x;
    assume
A6: x in Z;
    then
A7: -1 < x & x < 1 by A2,XXREAL_1:4;
    arcsin is_differentiable_in x & arcsin.x >0 by A2,A3,A6,FDIFF_1:9
,SIN_COS6:83;
    then diff(ln*arcsin,x) =diff(arcsin,x)/arcsin.x by TAYLOR_1:20
      .=(1 / sqrt(1-x^2))/arcsin.x by A7,SIN_COS6:83
      .=1 / (sqrt(1-x^2)*arcsin.x) by XCMPLX_1:78;
    hence thesis by A5,A6,FDIFF_1:def 7;
  end;
  hence thesis by A1,A4,FDIFF_1:9;
end;
