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 Th25:
  Z c= dom ((id Z)(#)((arcsin)*f)) & (for x st x in Z holds f.x=x/
a & f.x > -1 & f.x < 1) implies (id Z)(#)((arcsin)*f) is_differentiable_on Z &
for x st x in Z holds (((id Z)(#)((arcsin)*f))`|Z).x =arcsin.(x/a)+x/(a*sqrt(1-
  (x/a)^2))
proof
  assume that
A1: Z c= dom ((id Z)(#)((arcsin)*f)) and
A2: for x st x in Z holds f.x=x/a & f.x > -1 & f.x < 1;
A3: Z c= dom (id Z) /\ dom ((arcsin)*f) by A1,VALUED_1:def 4;
  then
A4: Z c= dom (id Z) by XBOOLE_1:18;
A5: Z c= dom ((arcsin)*f) by A3,XBOOLE_1:18;
  for x st x in Z holds f.x=(1/a)*x+0
  proof
    let x;
    assume x in Z;
    then f.x=x/a by A2;
    hence thesis by XCMPLX_1:99;
  end;
  then
A6: for x st x in Z holds f.x=(1/a)*x+0 & f.x > -1 & f.x < 1 by A2;
  then
A7: (arcsin)*f is_differentiable_on Z by A5,Th14;
A8: for x st x in Z holds (id Z).x = 1*x+0 by FUNCT_1:18;
  then
A9: id Z is_differentiable_on Z by A4,FDIFF_1:23;
A10: for x st x in Z holds (((arcsin)*f)`|Z).x=1/(a*sqrt(1-(x/a)^2))
  proof
    let x;
    assume x in Z;
    then (((arcsin)*f)`|Z).x=(1/a)/sqrt(1-((1/a)*x+0)^2) by A6,A5,Th14
      .=(1/a)/sqrt(1-(x/a)^2) by XCMPLX_1:99
      .=1/(a*sqrt(1-(x/a)^2)) by XCMPLX_1:78;
    hence thesis;
  end;
  for x st x in Z holds (((id Z)(#)((arcsin)*f))`|Z).x =arcsin.(x/a)+x/(a
  *sqrt(1-(x/a)^2))
  proof
    let x;
    assume
A11: x in Z;
    then
A12: ((arcsin)*f).x=arcsin.(f.x) by A5,FUNCT_1:12
      .=arcsin.(x/a) by A2,A11;
    (((id Z)(#)((arcsin)*f))`|Z).x =((arcsin)*f).x*diff((id Z),x) + ((id
    Z).x)*diff((arcsin)*f,x) by A1,A9,A7,A11,FDIFF_1:21
      .=((arcsin)*f).x*((id Z)`|Z).x + ((id Z).x)*diff((arcsin)*f,x) by A9,A11,
FDIFF_1:def 7
      .=((arcsin)*f).x*1+((id Z).x)*diff((arcsin)*f,x) by A4,A8,A11,FDIFF_1:23
      .=((arcsin)*f).x*1+((id Z).x)*(((arcsin)*f)`|Z).x by A7,A11,FDIFF_1:def 7
      .=((arcsin)*f).x+x*(((arcsin)*f)`|Z).x by A11,FUNCT_1:18
      .=arcsin.(x/a)+x*(1/(a*sqrt(1-(x/a)^2))) by A10,A11,A12
      .=arcsin.(x/a)+x/(a*sqrt(1-(x/a)^2)) by XCMPLX_1:99;
    hence thesis;
  end;
  hence thesis by A1,A9,A7,FDIFF_1:21;
end;
