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(#)(cos-sin),A) = (exp_R(#)cos).(upper_bound A)-(exp_R
  (#)cos).(lower_bound A)
proof
A1: dom (exp_R(#)cos) = REAL & [#]REAL is open Subset of REAL by FUNCT_2:def 1;
A2: dom (cos - sin) = REAL by FUNCT_2:def 1;
A3: for x being Element of REAL
    st x in dom ((exp_R(#)cos)`|REAL) holds ((exp_R(#)cos)`|REAL).x =
  (exp_R(#)(cos - sin)).x
  proof
    let x be Element of REAL;
    assume x in dom ((exp_R(#)cos)`|REAL);
    (exp_R(#)(cos - sin)).x = (exp_R.x) * ((cos - sin).x) by VALUED_1:5
      .= (exp_R.x) * (cos.x - sin.x) by A2,VALUED_1:13;
    hence thesis by A1,FDIFF_7:45;
  end;
A4: exp_R(#)(cos - sin) is_integrable_on A & (exp_R(#)(cos - sin))|A is
  bounded by Lm22;
A5: dom (exp_R(#)(cos - sin)) = dom exp_R /\ dom (cos - sin) by VALUED_1:def 4
    .= REAL /\ dom (cos - sin) by SIN_COS:47
    .= REAL by A2;
  (exp_R(#)cos) is_differentiable_on REAL by A1,FDIFF_7:45;
  then dom ((exp_R(#)cos)`|REAL) = dom (exp_R(#)(cos - sin)) by A5,
FDIFF_1:def 7;
  then ((exp_R(#)cos)`|REAL) = exp_R(#)(cos - sin) by A3,PARTFUN1:5;
  hence thesis by A4,A1,FDIFF_7:45,INTEGRA5:13;
end;
