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 Th88:
  Z c= dom ((arccot)*f) & (for x st x in Z holds f.x=r*x+s & f.x >
-1 & f.x < 1) implies (arccot)*f is_differentiable_on Z & for x st x in Z holds
  (((arccot)*f)`|Z).x = -r/(1+(r*x+s)^2)
proof
  assume that
A1: Z c= dom ((arccot)*f) and
A2: for x st x in Z holds f.x=r*x+s & 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;
A4: for x st x in Z holds f.x=r*x+s by A2;
  then
A5: f is_differentiable_on Z by A3,FDIFF_1:23;
A6: for x st x in Z holds (arccot)*f is_differentiable_in x
  proof
    let x;
    assume
A7: x in Z;
    then
A8: f.x > -1 by A2;
A9: f.x < 1 by A2,A7;
    f is_differentiable_in x by A5,A7,FDIFF_1:9;
    hence thesis by A8,A9,Th86;
  end;
  then
A10: (arccot)*f is_differentiable_on Z by A1,FDIFF_1:9;
  for x st x in Z holds (((arccot)*f)`|Z).x = -r/(1+(r*x+s)^2)
  proof
    let x;
    assume
A11: x in Z;
    then
A12: f.x > -1 by A2;
A13: f.x < 1 by A2,A11;
    f is_differentiable_in x by A5,A11,FDIFF_1:9;
    then diff((arccot)*f,x) = -diff(f,x)/(1+(f.x)^2) by A12,A13,Th86
      .= -(f`|Z).x/(1+(f.x)^2) by A5,A11,FDIFF_1:def 7
      .= -r/(1+(f.x)^2) by A4,A3,A11,FDIFF_1:23
      .= -r/(1+(r*x+s)^2) by A2,A11;
    hence thesis by A10,A11,FDIFF_1:def 7;
  end;
  hence thesis by A1,A6,FDIFF_1:9;
end;
