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(#)arcsin) implies r(#)arcsin
is_differentiable_on Z & for x st x in Z holds ((r(#)arcsin)`|Z).x = r / sqrt(1
  -x^2)
proof
  assume that
A1: Z c= ]. -1,1 .[ and
A2: Z c= dom (r(#)arcsin);
A3: arcsin is_differentiable_on Z by A1,FDIFF_1:26,SIN_COS6:83;
  for x st x in Z holds ((r(#)arcsin)`|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(#)arcsin)`|Z).x = r*diff(arcsin,x) by A2,A3,A4,FDIFF_1:20
      .= r*(1 / sqrt(1-x^2)) by A5,SIN_COS6:83
      .=r / sqrt(1-x^2) by XCMPLX_1:99;
    hence thesis;
  end;
  hence thesis by A2,A3,FDIFF_1:20;
end;
