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 (f(#)(arcsin)) & Z c= ]. -1,1 .[ & (for x st x in Z holds f.x
=a*x+b) implies f(#)(arcsin) is_differentiable_on Z & for x st x in Z holds ((f
  (#)(arcsin))`|Z).x =a*arcsin.x+(a*x+b)/sqrt(1-x^2)
proof
  assume that
A1: Z c= dom (f(#)(arcsin)) and
A2: Z c= ]. -1,1 .[ and
A3: for x st x in Z holds f.x=a*x+b;
  Z c= dom f /\ dom arcsin by A1,VALUED_1:def 4;
  then
A4: Z c= dom f by XBOOLE_1:18;
  then
A5: f is_differentiable_on Z by A3,FDIFF_1:23;
A6: arcsin is_differentiable_on Z by A2,FDIFF_1:26,SIN_COS6:83;
  for x st x in Z holds ((f(#)(arcsin))`|Z).x =a*arcsin.x+(a*x+b)/sqrt(1-x ^2)
  proof
    let x;
    assume
A7: x in Z;
    then
A8: -1 < x & x < 1 by A2,XXREAL_1:4;
    ((f(#)(arcsin))`|Z).x =(arcsin.x)*diff(f,x) + (f.x)*diff(arcsin,x) by A1,A5
,A6,A7,FDIFF_1:21
      .=(arcsin.x)*(f`|Z).x+ (f.x)*diff(arcsin,x) by A5,A7,FDIFF_1:def 7
      .=(arcsin.x)*a+ (f.x)*diff(arcsin,x) by A3,A4,A7,FDIFF_1:23
      .=(arcsin.x)*a+ (f.x)*(1 / sqrt(1-x^2)) by A8,SIN_COS6:83
      .=a*arcsin.x+ (a*x+b)*(1 / sqrt(1-x^2)) by A3,A7
      .=a*arcsin.x+(a*x+b) / sqrt(1-x^2) by XCMPLX_1:99;
    hence thesis;
  end;
  hence thesis by A1,A5,A6,FDIFF_1:21;
end;
