 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 f.x=a / sqrt(1-(a*x+b)^2)
 & f1.x=a*x+b & f1.x>-1 & f1.x<1) & Z c= dom ((arcsin)*f1)
 & Z = dom f & f|A is continuous implies
 integral(f,A)=((arcsin)*f1).(upper_bound A)-((arcsin)*f1).(lower_bound A)
proof
   assume
A1:A c= Z & (for x st x in Z holds f.x=a / sqrt(1-(a*x+b)^2)
   & f1.x=a*x+b & f1.x>-1 & f1.x<1) & Z c= dom ((arcsin)*f1)
   & Z = dom f & f|A is continuous;then
A2:f is_integrable_on A & f|A is bounded by INTEGRA5:10,11;
A3:for x st x in Z holds f1.x=a*x+b & f1.x>-1 & f1.x<1 by A1;
then A4:(arcsin)*f1 is_differentiable_on Z by A1,FDIFF_7:14;
A5:for x being Element of REAL
    st x in dom (((arcsin)*f1)`|Z) holds (((arcsin)*f1)`|Z).x=f.x
    proof
    let x be Element of REAL;
    assume x in dom(((arcsin)*f1)`|Z);then
A6:x in Z by A4,FDIFF_1:def 7; then
  (((arcsin)*f1)`|Z).x=a / sqrt(1-(a*x+b)^2) by A1,A3,FDIFF_7:14
  .=f.x by A1,A6;
   hence thesis;
   end;
  dom(((arcsin)*f1)`|Z)=dom f by A1,A4,FDIFF_1:def 7;
  then (((arcsin)*f1)`|Z)= f by A5,PARTFUN1:5;
  hence thesis by A1,A2,A4,INTEGRA5:13;
end;
