 reserve a,x for Real;
 reserve n for Element of NAT;
 reserve A for non empty closed_interval Subset of REAL;
 reserve f,h,f1,f2 for PartFunc of REAL,REAL;
 reserve Z for open Subset of REAL;

theorem Th50:
 Z c= dom (exp_R*arccot) & Z c= ]. -1,1 .[ implies
 -exp_R*arccot is_differentiable_on Z & for x st x in Z holds
 ((-exp_R*arccot)`|Z).x = exp_R.(arccot.x)/(1+x^2)
proof
   assume
A1:Z c= dom (exp_R*arccot) & Z c= ]. -1,1 .[;
then A2:Z c= dom (-exp_R*arccot) by VALUED_1:8;
A3:exp_R*arccot is_differentiable_on Z by A1,SIN_COS9:120;
then A4:(-1)(#)(exp_R*arccot) is_differentiable_on Z by A2,FDIFF_1:20;
  for x st x in Z holds ((-exp_R*arccot)`|Z).x = exp_R.(arccot.x)/(1+x^2)
    proof
      let x;
      assume
A5:x in Z;
A6:arccot is_differentiable_on Z by A1,SIN_COS9:82;then
A7:arccot is_differentiable_in x by A5,FDIFF_1:9;
A8:exp_R is_differentiable_in arccot.x by SIN_COS:65;
A9:exp_R*arccot is_differentiable_in x by A3,A5,FDIFF_1:9;
 ((-exp_R*arccot)`|Z).x=diff(-exp_R*arccot,x) by A4,A5,FDIFF_1:def 7
                      .=(-1)*(diff(exp_R*arccot,x)) by A9,FDIFF_1:15
                      .=(-1)*(diff(exp_R,arccot.x)*diff(arccot,x))
        by A7,A8,FDIFF_2:13
                      .=(-1)*(diff(exp_R,arccot.x)*((arccot)`|Z).x)
        by A5,A6,FDIFF_1:def 7
                      .=(-1)*(diff(exp_R,arccot.x)*(-1/(1+x^2)))
        by A5,A1,SIN_COS9:82
                      .=(-1)*(-diff(exp_R,arccot.x)*(1/(1+x^2)))
                      .=exp_R.(arccot.x)/(1+x^2) by SIN_COS:65;
      hence thesis;
   end;
   hence thesis by A2,A3,FDIFF_1:20;
end;
