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 ((1/2)(#)((arccos)*f)) & (for x st x in Z holds f.x=2*x & f.x
> -1 & f.x < 1) implies (1/2)(#)((arccos)*f) is_differentiable_on Z & for x st
  x in Z holds (((1/2)(#)((arccos)*f))`|Z).x=-1/sqrt(1-(2*x)^2)
proof
  assume that
A1: Z c= dom ((1/2)(#)((arccos)*f)) and
A2: for x st x in Z holds f.x=2*x & f.x > -1 & f.x < 1;
A3: Z c= dom ((arccos)*f) & for x st x in Z holds f.x=2*x+0 & f.x > -1 & f.x
  < 1 by A1,A2,VALUED_1:def 5;
  then
A4: (arccos)*f is_differentiable_on Z by Th15;
  for x st x in Z holds (((1/2)(#)((arccos)*f))`|Z).x=-1/sqrt(1-(2*x)^2)
  proof
    let x;
    assume
A5: x in Z;
    then (((1/2)(#)((arccos)*f))`|Z).x =(1/2)*diff(((arccos)*f),x) by A1,A4,
FDIFF_1:20
      .=(1/2)*(((arccos)*f)`|Z).x by A4,A5,FDIFF_1:def 7
      .=(1/2)*(-2 / sqrt(1-(2*x+0)^2)) by A3,A5,Th15
      .=-(1/2)*(2 / sqrt(1-(2*x)^2))
      .=-((1/2)*2)/sqrt(1-(2*x)^2) by XCMPLX_1:74
      .=-1/sqrt(1-(2*x)^2);
    hence thesis;
  end;
  hence thesis by A1,A4,FDIFF_1:20;
end;
