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(#)exp_R,A) =1/2*((exp_R.(upper_bound A))^2 -((exp_R).(
  lower_bound A))^2)
proof
  exp_R|A is continuous;
  then
A1: exp_R`|REAL is_integrable_on A by Lm8,Th32,INTEGRA5:11;
  (exp_R`|REAL)|A is bounded & [#]REAL is open Subset of REAL by Lm8,Th32,
INTEGRA5:10;
  then
  integral(exp_R(#)exp_R,A) =(exp_R).(upper_bound A)*exp_R.(upper_bound A)
- (exp_R).(lower_bound A)*exp_R.(lower_bound A) -integral((exp_R)(#)(exp_R),A)
  by A1,Th32,INTEGRA5:21,SIN_COS:66
    .= (exp_R.(upper_bound A))^2 - ((exp_R).(lower_bound A))*exp_R.(
  lower_bound A) -integral(exp_R(#)exp_R,A)
    .= (exp_R.(upper_bound A))^2 - ((exp_R).(lower_bound A))^2 -integral(
  exp_R(#)exp_R,A);
  hence thesis;
end;
