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 ((id Z)(#)((arcsin)*f3)+( #R (1/2))*f) & f=f1-f2 & f2=#Z 2 &
(for x st x in Z holds f1.x=a^2 & f.x >0 & f3.x=x/a & f3.x > -1 & f3.x < 1 & x
<>0 & a>0) implies (id Z)(#)((arcsin)*f3)+( #R (1/2))*f is_differentiable_on Z
& for x st x in Z holds (((id Z)(#)((arcsin)*f3)+( #R (1/2))*f)`|Z).x = arcsin.
  (x/a)
proof
  assume that
A1: Z c= dom ((id Z)(#)((arcsin)*f3)+( #R (1/2))*f) and
A2: f=f1-f2 & f2=#Z 2 and
A3: for x st x in Z holds f1.x=a^2 & f.x >0 & f3.x=x/a & f3.x > -1 & f3
  .x < 1 & x<>0 & a>0;
A4: Z c= dom ((id Z)(#)((arcsin)*f3)) /\ dom (( #R (1/2))*f) by A1,
VALUED_1:def 1;
  then
A5: Z c= dom (( #R (1/2))*f) by XBOOLE_1:18;
A6: for x st x in Z holds f1.x=a^2 & f.x >0 by A3;
  then
A7: ( #R (1/2))*f is_differentiable_on Z by A2,A5,Th27;
A8: for x st x in Z holds f3.x=x/a & f3.x > -1 & f3.x < 1 by A3;
A9: Z c= dom ((id Z)(#)((arcsin)*f3)) by A4,XBOOLE_1:18;
  then
A10: (id Z)(#)((arcsin)*f3) is_differentiable_on Z by A8,Th25;
A11: for x st x in Z holds (a^2-x #Z 2) #R (-1/2)=1/(a*sqrt(1-(x/a)^2))
  proof
    let x;
    assume
A12: x in Z;
    then
A13: f3.x=x/a by A3;
    f3.x < 1 by A3,A12;
    then
A14: 1-f3.x >1-1 by XREAL_1:10;
    f3.x > -1 by A3,A12;
    then 1+f3.x>1+(-1) by XREAL_1:6;
    then
A15: (1+f3.x)*(1-f3.x)>0 by A14,XREAL_1:129;
A16: a>0 by A3,A12;
    then
A17: a^2>0 by SQUARE_1:12;
    1/(a*sqrt(1-(x/a)^2)) =1/((sqrt a^2)* sqrt (1-(x/a)^2)) by A16,SQUARE_1:22
      .=1/sqrt ((a^2)*(1-(x/a)^2)) by A17,A13,A15,SQUARE_1:29
      .=(a^2*1 - (a *(x/a))^2) #R (-1/2) by A17,A13,A15,Th3,XREAL_1:129
      .=(a^2 - (x*a/a)^2) #R (-1/2) by XCMPLX_1:74
      .=(a^2 - (x*(a/a))^2) #R (-1/2) by XCMPLX_1:74
      .=(a^2 - (x*1)^2) #R (-1/2) by A16,XCMPLX_1:60
      .=(a^2-x #Z 2) #R (-1/2) by Th1;
    hence thesis;
  end;
  for x st x in Z holds (((id Z)(#)((arcsin)*f3)+( #R (1/2))*f)`|Z).x =
  arcsin.(x/a)
  proof
    let x;
    assume
A18: x in Z;
    then (((id Z)(#)((arcsin)*f3)+( #R (1/2))*f)`|Z).x = diff((id Z)(#)((
    arcsin)*f3),x) + diff(( #R (1/2))*f,x) by A1,A10,A7,FDIFF_1:18
      .=(((id Z)(#)((arcsin)*f3))`|Z).x+ diff(( #R (1/2))*f,x) by A10,A18,
FDIFF_1:def 7
      .=(((id Z)(#)((arcsin)*f3))`|Z).x+ ((( #R (1/2))*f)`|Z).x by A7,A18,
FDIFF_1:def 7
      .=arcsin.(x/a)+x/(a*sqrt(1-(x/a)^2)) + ((( #R (1/2))*f)`|Z).x by A9,A8
,A18,Th25
      .=arcsin.(x/a)+x/(a*sqrt(1-(x/a)^2)) + -x*(a^2-x #Z 2) #R (-1/2) by A2,A5
,A6,A18,Th27
      .=arcsin.(x/a)+x/(a*sqrt(1-(x/a)^2)) -x*(a^2-x #Z 2) #R (-1/2)
      .=arcsin.(x/a)+x/(a*sqrt(1-(x/a)^2)) -x*(1/(a*sqrt(1-(x/a)^2))) by A11
,A18
      .=arcsin.(x/a)+x/(a*sqrt(1-(x/a)^2)) -x/(a*sqrt(1-(x/a)^2)) by
XCMPLX_1:99
      .=arcsin.(x/a);
    hence thesis;
  end;
  hence thesis by A1,A10,A7,FDIFF_1:18;
end;
