reserve i,n,m for Nat,
        r,s for Real,
        A for non empty closed_interval Subset of REAL;

theorem Th3:
  for Z be open Subset of REAL holds
    arctan is_differentiable_on Z &
    for r st r in Z holds ((arctan)`|Z).r = 1/(1+r^2)
proof
  let Z be open Subset of REAL;
A1: dom arctan = REAL by FUNCT_2:def 1;
A2: arctan is_differentiable_on Z by A1, FDIFF_1:28;
  r in Z implies ((arctan)`|Z).r = 1/(1+r^2)
  proof
    assume r in Z;
    hence ((arctan)`|Z).r = diff(arctan,r) by A2,FDIFF_1:def 7
                         .= 1/(1+r^2) by Th2;
  end;
  hence thesis by A1,FDIFF_1:28;
end;
