 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 & (for x st x in Z holds f1.x=1)
 & Z c= ]. -1,1 .[ & Z = dom f & f=arccos-(id Z)/(( #R (1/2))*(f1-#Z 2))
 implies
 integral(f,A)
 =((id Z)(#)(arccos)).(upper_bound A)-((id Z)(#)(arccos)).(lower_bound A)
proof
  assume
A1:A c= Z & (for x st x in Z holds f1.x=1) & Z c= ]. -1,1 .[ & Z = dom f
   & f=arccos-(id Z)/(( #R (1/2))*(f1-#Z 2));
then
Z = dom arccos /\ dom ((id Z)/(( #R (1/2))*(f1-#Z 2))) by VALUED_1:12;
then A2:Z c= dom arccos & Z c= dom ((id Z)/(( #R (1/2))*(f1-#Z 2)))
   by XBOOLE_1:18;
A3:Z = dom id Z;
Z c= dom id Z /\ dom arccos by A2,XBOOLE_1:19;
then A4:Z c= dom ((id Z)(#)(arccos)) by VALUED_1:def 4;
A5:arccos is_differentiable_on Z by A1,FDIFF_1:26,SIN_COS6:106;
   for x st x in Z holds (id Z).x = 1*x+0 by FUNCT_1:18;then
A6:id Z is_differentiable_on Z by A3,FDIFF_1:23;
Z c= dom id Z /\ (dom (( #R (1/2))*(f1-#Z 2)) \ (( #R (1/2))*(f1-#Z 2))"{0})
   by A2,RFUNCT_1:def 1;
then Z c= dom (( #R (1/2))*(f1-#Z 2)) \ (( #R (1/2))*(f1-#Z 2))"{0}
    by XBOOLE_1:18;
then A7:Z c= dom ((( #R (1/2))*(f1-#Z 2))^) by RFUNCT_1:def 2;
   dom ((( #R (1/2))*(f1-#Z 2))^) c= dom (( #R (1/2))*(f1-#Z 2))
   by RFUNCT_1:1; then
A8:Z c= dom (( #R (1/2))*(f1-#Z 2)) by A7;
    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 A8,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 A8,FDIFF_7:22;
x in Z implies (( #R (1/2))*(f1-#Z 2)).x<>0 by A7,RFUNCT_1:3;
then (id Z)/(( #R (1/2))*(f1-#Z 2)) is_differentiable_on Z
    by A6,A11,FDIFF_2:21;
    then f|Z is continuous by A1,A5,FDIFF_1:19,25;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:(id Z)(#)(arccos) is_differentiable_on Z by A1,A4,FDIFF_7:17;
A14:for x st x in Z holds f.x=arccos.x-x/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 A8,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;
   (arccos-(id Z)/(( #R (1/2))*(f1-#Z 2))).x
   =arccos.x-((id Z)/(( #R (1/2))*(f1-#Z 2))).x by A1,A15,VALUED_1:13
  .=arccos.x-(id Z).x/(( #R (1/2))*(f1-#Z 2)).x by A2,A15,RFUNCT_1:def 1
  .=arccos.x-x/(( #R (1/2))*(f1-#Z 2)).x by A15,FUNCT_1:18
  .=arccos.x-x/(( #R (1/2)).((f1-#Z 2).x)) by A8,A15,FUNCT_1:12
  .=arccos.x-x/(((f1-#Z 2).x) #R (1/2)) by A17,TAYLOR_1:def 4
  .=arccos.x-x/((f1.x-(( #Z 2).x)) #R (1/2)) by A16,VALUED_1:13
  .=arccos.x-x/((f1.x-(x #Z 2)) #R (1/2)) by TAYLOR_1:def 1
  .=arccos.x-x/((f1.x-x^2) #R (1/2)) by FDIFF_7:1
  .=arccos.x-x/((1-x^2) #R (1/2)) by A1,A15
  .=arccos.x-x/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 (((id Z)(#)(arccos))`|Z) holds
   (((id Z)(#)(arccos))`|Z).x=f.x
    proof
    let x be Element of REAL;
    assume x in dom(((id Z)(#)(arccos))`|Z);then
A20:x in Z by A13,FDIFF_1:def 7; then
  (((id Z)(#)(arccos))`|Z).x=arccos.x-x/sqrt(1-x^2) by A1,A4,FDIFF_7:17
   .=f.x by A14,A20;
   hence thesis;
   end;
  dom(((id Z)(#)(arccos))`|Z)=dom f by A1,A13,FDIFF_1:def 7;
  then(((id Z)(#)(arccos))`|Z)= f by A19,PARTFUN1:5;
  hence thesis by A1,A12,A4,FDIFF_7:17,INTEGRA5:13;
end;
