reserve x for Real,

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

     g for PartFunc of REAL,REAL;

theorem
  Z c= dom (((2/3)(#)(( #R (3/2))*arccot))) & Z c= ].-1,1.[ implies (2/3
)(#)(( #R (3/2))*arccot) is_differentiable_on Z & for x st x in Z holds (((2/3)
  (#)(( #R (3/2))*arccot))`|Z).x = -(arccot.x) #R (1/2)/(1+x^2)
proof
  assume that
A1: Z c= dom (((2/3)(#)(( #R (3/2))*arccot))) and
A2: Z c= ].-1,1.[;
A3: for x st x in Z holds arccot.x > 0
  proof
    let x;
    ].-1,1.[ c= [.-1,1.] by XXREAL_1:25;
    then
A4: Z c= [.-1,1.] by A2,XBOOLE_1:1;
    assume x in Z;
    then x in [.-1,1.] by A4;
    then arccot.x in arccot.:[.-1,1.] by FUNCT_1:def 6,SIN_COS9:24;
    then arccot.x in [.PI/4,3/4*PI.] by RELAT_1:115,SIN_COS9:56;
    then arccot.x >= PI/4 by XXREAL_1:1;
    then
A5: arccot.x + 0 > PI/4 + (-PI/4) by XREAL_1:8;
    assume arccot.x <= 0;
    hence contradiction by A5;
  end;
A6: for x st x in Z holds ( #R (3/2))*arccot is_differentiable_in x
  proof
    let x;
    assume
A7: x in Z;
    then
A8: arccot.x > 0 by A3;
    arccot is_differentiable_on Z by A2,SIN_COS9:82;
    then arccot is_differentiable_in x by A7,FDIFF_1:9;
    hence thesis by A8,TAYLOR_1:22;
  end;
  Z c= dom (( #R (3/2))*arccot) by A1,VALUED_1:def 5;
  then
A9: ( #R (3/2))*arccot is_differentiable_on Z by A6,FDIFF_1:9;
  for x st x in Z holds (((2/3)(#)(( #R (3/2))*arccot))`|Z).x = -(arccot.
  x) #R (1/2)/(1+x^2)
  proof
    let x;
    assume
A10: x in Z;
    then
A11: arccot.x > 0 by A3;
A12: arccot is_differentiable_on Z by A2,SIN_COS9:82;
    then
A13: arccot is_differentiable_in x by A10,FDIFF_1:9;
    (((2/3)(#)(( #R (3/2))*arccot))`|Z).x = (2/3)*diff((( #R (3/2))*
    arccot),x) by A1,A9,A10,FDIFF_1:20
      .= (2/3)*((3/2)*((arccot.x) #R (3/2-1))*diff(arccot,x)) by A13,A11,
TAYLOR_1:22
      .= (2/3)*((3/2)*((arccot.x) #R (3/2-1))*((arccot)`|Z).x) by A10,A12,
FDIFF_1:def 7
      .= (2/3)*((3/2)*((arccot.x) #R (3/2-1))*(-1/(1+x^2))) by A2,A10,
SIN_COS9:82
      .= -((arccot.x) #R (1/2))/(1+x^2);
    hence thesis;
  end;
  hence thesis by A1,A9,FDIFF_1:20;
end;
