reserve y for set,
  x,a,b for Real,
  n for Element of NAT,
  Z for open Subset of REAL,
  f,f1,f2,g for PartFunc of REAL,REAL;

theorem
  Z c= dom (sin+( #R (1/2))) implies sin+( #R (1/2))
is_differentiable_on Z & for x st x in Z holds ((sin+( #R (1/2)))`|Z).x =cos.x+
  (1/2)* x #R (-1/2)
proof
  assume
A1: Z c= dom (sin+( #R (1/2)));
  then Z c= dom (( #R (1/2))) /\ dom sin by VALUED_1:def 1;
  then
A2: Z c= dom (( #R (1/2))) by XBOOLE_1:18;
  then
A3: ( #R (1/2)) is_differentiable_on Z by Lm3;
A4: sin is_differentiable_on Z by FDIFF_1:26,SIN_COS:68;
  now
    let x;
    assume
A5: x in Z;
    then
    ((sin+( #R (1/2)))`|Z).x = diff(sin,x) + diff(( #R (1/2)),x) by A1,A3,A4,
FDIFF_1:18
      .=cos.x + diff(( #R (1/2)),x) by SIN_COS:64
      .=cos.x + ((( #R (1/2)))`|Z).x by A3,A5,FDIFF_1:def 7
      .=cos.x +(1/2)*x #R (-1/2) by A2,A5,Lm3;
    hence ((sin+( #R (1/2)))`|Z).x =cos.x+(1/2)*x #R (-1/2);
  end;
  hence thesis by A1,A3,A4,FDIFF_1:18;
end;
