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 Th7:
  f is_differentiable_in x & f.x > -1 & f.x < 1 implies (arccos)*f
  is_differentiable_in x & diff((arccos)*f,x) = -diff(f,x)/sqrt(1-(f.x)^2)
proof
  assume that
A1: f is_differentiable_in x and
A2: f.x > -1 & f.x < 1;
  f.x in ]. -1,1 .[ by A2;
  then
A3: arccos is_differentiable_in f.x by FDIFF_1:9,SIN_COS6:106;
  then diff(arccos*f,x) = diff(arccos,f.x)*diff(f,x) by A1,FDIFF_2:13
    .=(-1 / sqrt(1-(f.x)^2))*diff(f,x) by A2,SIN_COS6:106
    .=-diff(f,x)*(1 / sqrt(1-(f.x)^2))
    .=-diff(f,x)/sqrt(1-(f.x)^2) by XCMPLX_1:99;
  hence thesis by A1,A3,FDIFF_2:13;
end;
