 reserve a,b,x,r for Real;
 reserve y for set;
 reserve n for Element of NAT;
 reserve A for non empty closed_interval Subset of REAL;
 reserve f,g,f1,f2,g1,g2 for PartFunc of REAL,REAL;
 reserve Z for open Subset of REAL;

theorem
 A c= Z & Z c= ]. -1,1 .[  & (for x st x in Z holds f1.x=1)
 & Z = dom f & Z c= dom (( #Z n)*arcsin) & 1<n &
 f=n(#)(( #Z (n-1))*arcsin)/(( #R (1/2))*(f1-#Z 2))
 implies integral(f,A) = (( #Z n)*arcsin).(upper_bound A)
                        -(( #Z n)*arcsin).(lower_bound A)
proof
   assume
A1:A c= Z & Z c= ]. -1,1 .[  & (for x st x in Z holds f1.x=1)
   & Z = dom f & Z c= dom (( #Z n)*arcsin) & 1<n &
   f=n(#)(( #Z (n-1))*arcsin)/(( #R (1/2))*(f1-#Z 2));
then Z = dom (n(#)(( #Z (n-1))*(arcsin))) /\
      (dom (( #R (1/2))*(f1-#Z 2)) \ (( #R (1/2))*(f1-#Z 2))"{0})
      by RFUNCT_1:def 1;
then A2:Z c= dom (n(#)(( #Z (n-1))*arcsin)) &
   Z c= dom (( #R (1/2))*(f1-#Z 2)) \ (( #R (1/2))*(f1-#Z 2))"{0}
   by XBOOLE_1:18;
then A3:Z c= dom (( #Z (n-1))*arcsin) by VALUED_1:def 5;
A4:Z c= dom ((( #R (1/2))*(f1-#Z 2))^) by A2,RFUNCT_1:def 2;
   dom ((( #R (1/2))*(f1-#Z 2))^) c= dom (( #R (1/2))*(f1-#Z 2))
   by RFUNCT_1:1; then
A5:Z c= dom (( #R (1/2))*(f1-#Z 2)) by A4;
for x st x in Z holds ( #Z (n-1))*arcsin is_differentiable_in x
   proof
     let x;
     assume x in Z; then
A6:  arcsin is_differentiable_in x by A1,FDIFF_1:9,SIN_COS6:83;
     consider m being Nat such that
A7:  n = m + 1 by A1,NAT_1:6;
     thus thesis by A6,A7,TAYLOR_1:3;
   end;
then ( #Z (n-1))*arcsin is_differentiable_on Z by A3,FDIFF_1:9;
then A8:n(#)(( #Z (n-1))*arcsin) is_differentiable_on Z by A2,FDIFF_1:20;
    set f2=#Z 2;
for x st x in Z holds (f1-#Z 2).x >0
   proof
   let x;
   assume
A9:x in Z; then
   -1 < x & x < 1 by A1,XXREAL_1:4; then
   0 < 1+x & 0 < 1-x by XREAL_1:50,148; then
A10:0 < (1+x)*(1-x) by XREAL_1:129;
for x st x in Z holds x in dom (f1-f2) by A5,FUNCT_1:11;
     then (f1-f2).x = f1.x - f2.x by A9,VALUED_1:13
             .=f1.x - (x #Z (1+1)) by TAYLOR_1:def 1
             .=f1.x - ((x #Z 1)*(x #Z 1)) by TAYLOR_1:1
             .=f1.x - (x*(x #Z 1)) by PREPOWER:35
             .=f1.x - x*x by PREPOWER:35
             .=1 - x*x by A1,A9;
     hence thesis by A10;
    end;
then for x st x in Z holds f1.x=1 & (f1-#Z 2).x >0 by A1;
then A11:(( #R (1/2))*(f1-#Z 2)) is_differentiable_on Z by A5,FDIFF_7:22;
x in Z implies (( #R (1/2))*(f1-#Z 2)).x<>0 by A4,RFUNCT_1:3;
then f is_differentiable_on Z by A1,A8,A11,FDIFF_2:21;
   then f|Z is continuous by FDIFF_1:25;then
f|A is continuous by A1,FCONT_1:16;
then A12:f is_integrable_on A & f|A is bounded by A1,INTEGRA5:10,11;
A13:( #Z n)*arcsin is_differentiable_on Z by A1,FDIFF_7:10;
A14:for x st x in Z holds f.x=n*(arcsin.x) #Z (n-1) / sqrt(1-x^2)
    proof
    let x;
    assume
A15:x in Z;
then
A16:x in dom (f1-#Z 2) & (f1-#Z 2).x in dom ( #R (1/2)) by A5,FUNCT_1:11;
then A17:(f1-#Z 2).x in right_open_halfline(0) by TAYLOR_1:def 4;
   -1 < x & x < 1 by A1,A15,XXREAL_1:4; then
   0 < 1+x & 0 < 1-x by XREAL_1:50,148; then
A18:0 < (1+x)*(1-x) by XREAL_1:129;
   (n(#)(( #Z (n-1))*arcsin)/(( #R (1/2))*(f1-#Z 2))).x
  =(n(#)(( #Z (n-1))*arcsin)).x/(( #R (1/2))*(f1-#Z 2)).x
    by A1,A15,RFUNCT_1:def 1
 .=n*(( #Z (n-1))*arcsin).x/(( #R (1/2))*(f1-#Z 2)).x by VALUED_1:6
 .=n*(( #Z (n-1)).(arcsin.x))/(( #R (1/2))*(f1-#Z 2)).x
    by A3,A15,FUNCT_1:12
 .=n*(arcsin.x) #Z (n-1) / (( #R (1/2))*(f1-#Z 2)).x by TAYLOR_1:def 1
 .=n*(arcsin.x) #Z (n-1) / (( #R (1/2)).((f1-#Z 2).x)) by A5,A15,FUNCT_1:12
 .=n*(arcsin.x) #Z (n-1) / (((f1-#Z 2).x) #R (1/2)) by A17,TAYLOR_1:def 4
 .=n*(arcsin.x) #Z (n-1) / ((f1.x-(( #Z 2).x)) #R (1/2))
    by A16,VALUED_1:13
 .=n*(arcsin.x) #Z (n-1) / ((f1.x-(x #Z 2)) #R (1/2)) by TAYLOR_1:def 1
 .=n*(arcsin.x) #Z (n-1) / ((f1.x-x^2) #R (1/2)) by FDIFF_7:1
 .=n*(arcsin.x) #Z (n-1) / ((1-x^2) #R (1/2)) by A1,A15
 .=n*(arcsin.x) #Z (n-1) / sqrt(1-x^2) by A18,FDIFF_7:2;
    hence thesis by A1;
    end;
A19:for x being Element of REAL st x in dom ((( #Z n)*(arcsin))`|Z) holds
    ((( #Z n)*(arcsin))`|Z).x=f.x
     proof
     let x be Element of REAL;
     assume x in dom ((( #Z n)*(arcsin))`|Z);then
A20:x in Z by A13,FDIFF_1:def 7; then
  ((( #Z n)*(arcsin))`|Z).x=n*(arcsin.x) #Z (n-1) / sqrt(1-x^2)
   by A1,FDIFF_7:10
   .=f.x by A14,A20;
   hence thesis;
   end;
  dom ((( #Z n)*(arcsin))`|Z)=dom f by A1,A13,FDIFF_1:def 7;
  then ((( #Z n)*(arcsin))`|Z)= f by A19,PARTFUN1:5;
  hence thesis by A1,A12,FDIFF_7:10,INTEGRA5:13;
end;
