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 Th15:
  Z c= dom ((arccos)*f) & (for x st x in Z holds f.x=a*x+b & f.x >
-1 & f.x < 1) implies (arccos)*f is_differentiable_on Z & for x st x in Z holds
  (((arccos)*f)`|Z).x=-a / sqrt(1-(a*x+b)^2)
proof
  assume that
A1: Z c= dom ((arccos)*f) and
A2: for x st x in Z holds f.x=a*x+b & f.x > -1 & f.x < 1;
  for y being object st y in Z holds y in dom f by A1,FUNCT_1:11;
  then
A3: Z c= dom f by TARSKI:def 3;
A4: for x st x in Z holds f.x=a*x+b by A2;
  then
A5: f is_differentiable_on Z by A3,FDIFF_1:23;
A6: for x st x in Z holds (arccos)*f is_differentiable_in x
  proof
    let x;
    assume
A7: x in Z;
    then
A8: f.x < 1 by A2;
    f is_differentiable_in x & f.x > -1 by A2,A5,A7,FDIFF_1:9;
    hence thesis by A8,Th7;
  end;
  then
A9: (arccos)*f is_differentiable_on Z by A1,FDIFF_1:9;
  for x st x in Z holds (((arccos)*f)`|Z).x=-a / sqrt(1-(a*x+b)^2)
  proof
    let x;
    assume
A10: x in Z;
    then
A11: f.x < 1 by A2;
    f is_differentiable_in x & f.x > -1 by A2,A5,A10,FDIFF_1:9;
    then diff((arccos)*f,x) = -diff(f,x)/sqrt(1-(f.x)^2) by A11,Th7
      .=-(f`|Z).x/sqrt(1-(f.x)^2) by A5,A10,FDIFF_1:def 7
      .=-a/sqrt(1-(f.x)^2) by A4,A3,A10,FDIFF_1:23
      .=-a / sqrt(1-(a*x+b)^2) by A2,A10;
    hence thesis by A9,A10,FDIFF_1:def 7;
  end;
  hence thesis by A1,A6,FDIFF_1:9;
end;
