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 Th90:
  integral(sin(#)cos,A) = 1/2*(cos.(lower_bound A)*cos.(
  lower_bound A)- cos.(upper_bound A)*cos.(upper_bound A))
proof
A1: [#]REAL is open Subset of REAL;
  sin|A is continuous;
  then
A2: (-cos)`|REAL is_integrable_on A by Lm5,Th29,INTEGRA5:11;
  (-sin)|A is continuous;
  then
A3: cos`|REAL is_integrable_on A by Lm7,Th28,INTEGRA5:11;
  ((-cos)`|REAL)|A is bounded & (cos`|REAL)|A is bounded by Lm5,Lm7,Th28,Th29,
INTEGRA5:10;
  then
  integral(sin(#)cos,A) =(-cos).(upper_bound A)*cos.(upper_bound A)- (-cos
  ).(lower_bound A)*cos.(lower_bound A) -integral((-cos)(#)(-sin),A) by A2,A3
,A1,Th26,Th28,Th29,INTEGRA5:21,SIN_COS:67
    .=(-cos).(upper_bound A)*cos.(upper_bound A)- (-cos).(lower_bound A)*cos
  .(lower_bound A) -integral(sin(#)cos,A) by Lm4
    .=(-cos.(upper_bound A))*cos.(upper_bound A)- (-cos).(lower_bound A)*cos
  .(lower_bound A) -integral(sin(#)cos,A) by VALUED_1:8
    .=(-cos.(upper_bound A))*cos.(upper_bound A)- (-cos.(lower_bound A))*cos
  .(lower_bound A) -integral(sin(#)cos,A) by VALUED_1:8
    .=cos.(lower_bound A)*cos.(lower_bound A)- cos.(upper_bound A)*cos.(
  upper_bound A) -integral(sin(#)cos,A);
  hence thesis;
end;
