reserve x,x0, r, s, h for Real,

  n for Element of NAT,
  rr, y for set,
  Z for open Subset of REAL,

  f, f1, f2 for PartFunc of REAL,REAL;

theorem Th105:
  Z c= dom ((id Z)(#)((arctan)*f)) & (for x st x in Z holds f.x=x
/r & f.x > -1 & f.x < 1) implies (id Z)(#)((arctan)*f) is_differentiable_on Z &
for x st x in Z holds (((id Z)(#)((arctan)*f))`|Z).x = arctan.(x/r)+x/(r*(1+(x/
  r)^2))
proof
  assume that
A1: Z c= dom ((id Z)(#)((arctan)*f)) and
A2: for x st x in Z holds f.x=x/r & f.x > -1 & f.x < 1;
A3: Z c= dom (id Z) /\ dom ((arctan)*f) by A1,VALUED_1:def 4;
  then
A4: Z c= dom (id Z) by XBOOLE_1:18;
A5: Z c= dom ((arctan)*f) by A3,XBOOLE_1:18;
  for x st x in Z holds f.x=(1/r)*x+0
  proof
    let x;
    assume x in Z;
    then f.x=x/r by A2;
    hence thesis;
  end;
  then
A6: for x st x in Z holds f.x=(1/r)*x+0 & f.x > -1 & f.x < 1 by A2;
  then
A7: (arctan)*f is_differentiable_on Z by A5,Th87;
A8: for x st x in Z holds (id Z).x = 1*x+0 by FUNCT_1:18;
  then
A9: id Z is_differentiable_on Z by A4,FDIFF_1:23;
A10: for x st x in Z holds (((arctan)*f)`|Z).x = 1/(r*(1+(x/r)^2))
  proof
    let x;
    assume x in Z;
    then (((arctan)*f)`|Z).x = (1/r)/(1+((1/r)*x+0)^2) by A6,A5,Th87
      .= 1/(r*(1+(x/r)^2)) by XCMPLX_1:78;
    hence thesis;
  end;
  for x st x in Z holds (((id Z)(#)((arctan)*f))`|Z).x = arctan.(x/r)+x/(
  r*(1+(x/r)^2))
  proof
    let x;
    assume
A11: x in Z;
    then
A12: ((arctan)*f).x = arctan.(f.x) by A5,FUNCT_1:12
      .= arctan.(x/r) by A2,A11;
    (((id Z)(#)((arctan)*f))`|Z).x = ((arctan)*f).x*diff((id Z),x) + ((id
    Z).x)*diff((arctan)*f,x) by A1,A9,A7,A11,FDIFF_1:21
      .= ((arctan)*f).x*((id Z)`|Z).x + ((id Z).x)*diff((arctan)*f,x) by A9,A11
,FDIFF_1:def 7
      .= ((arctan)*f).x*1 + ((id Z).x)*diff((arctan)*f,x) by A4,A8,A11,
FDIFF_1:23
      .= ((arctan)*f).x*1 + ((id Z).x)*(((arctan)*f)`|Z).x by A7,A11,
FDIFF_1:def 7
      .= ((arctan)*f).x + x*(((arctan)*f)`|Z).x by A11,FUNCT_1:18
      .= arctan.(x/r) + x*(1/(r*(1+(x/r)^2))) by A10,A11,A12
      .= arctan.(x/r) + x/(r*(1+(x/r)^2));
    hence thesis;
  end;
  hence thesis by A1,A9,A7,FDIFF_1:21;
end;
