reserve f,f1,f2,g for PartFunc of REAL,REAL;
reserve A for non empty closed_interval Subset of REAL;
reserve p,r,x,x0 for Real;
reserve n for Element of NAT;
reserve Z for open Subset of REAL;

theorem
  integral(exp_R(#)(sin+cos),A) = (exp_R(#)sin).(upper_bound A)-(exp_R
  (#)sin).(lower_bound A)
proof
A1: dom (sin+cos) = REAL by FUNCT_2:def 1;
A2: dom (exp_R(#)(sin+cos)) = dom exp_R /\ dom (sin+cos) by VALUED_1:def 4
    .= REAL /\ dom (sin+cos) by SIN_COS:47
    .= REAL by A1;
A3: exp_R(#)(sin + cos) is_integrable_on A & (exp_R(#)(sin +cos))|A is
  bounded by Lm21;
A4: dom (exp_R(#)sin) = REAL & [#]REAL is open Subset of REAL by FUNCT_2:def 1;
A5: for x being Element of REAL
    st x in dom ((exp_R(#)sin)`|REAL) holds ((exp_R(#)sin)`|REAL).x =
  (exp_R(#)(sin + cos)).x
  proof
    let x be Element of REAL;
     reconsider xx=x as Real;
    assume x in dom ((exp_R(#)sin)`|REAL);
    (exp_R(#)(sin + cos)).x = (exp_R.x) * ((sin + cos).x) by VALUED_1:5
      .= (exp_R.x) * (sin.xx + cos.xx) by VALUED_1:1;
    hence thesis by A4,FDIFF_7:44;
  end;
  (exp_R(#)sin) is_differentiable_on REAL by A4,FDIFF_7:44;
  then dom ((exp_R(#)sin)`|REAL) = dom (exp_R(#)(sin + cos)) by A2,
FDIFF_1:def 7;
  then ((exp_R(#)sin)`|REAL) = exp_R(#)(sin + cos) by A5,PARTFUN1:5;
  hence thesis by A3,A4,FDIFF_7:44,INTEGRA5:13;
end;
