reserve r,p,x for Real;
reserve n for Element of NAT;
reserve A for non empty closed_interval Subset of REAL;
reserve Z for open Subset of REAL;

theorem
  r<>0 implies integral(exp_R*(AffineMap(r,0)),A) = ((1/r)(#)(exp_R*(
  AffineMap(r,0)))).(upper_bound A) -((1/r)(#)(exp_R*(AffineMap(r,0)))).(
  lower_bound A)
proof
A1: dom (exp_R*(AffineMap(r,0))) = REAL by FUNCT_2:def 1;
  assume
A2: r<>0;
  then (1/r)(#)(exp_R*(AffineMap(r,0))) is_differentiable_on REAL by Th2;
  then
A3: dom (((1/r)(#)(exp_R*(AffineMap(r,0))))`|REAL) = dom (exp_R*(AffineMap(
  r,0))) by A1,FDIFF_1:def 7;
  (exp_R*AffineMap(r,0))|A is continuous;
  then
A4: (exp_R*AffineMap(r,0)) is_integrable_on A by A1,INTEGRA5:11;
  for x being Element of REAL
   st x in dom (((1/r)(#)(exp_R*(AffineMap(r,0))))`|REAL) holds (((1/
r)(#)(exp_R*(AffineMap(r,0))))`|REAL).x = (exp_R*(AffineMap(r,0))).x by A2,Th2;
  then
A5: (((1/r)(#)(exp_R*(AffineMap(r,0))))`|REAL) = exp_R*(AffineMap(r,0)) by A3,
PARTFUN1:5;
  (exp_R*AffineMap(r,0))|A is bounded by A1,INTEGRA5:10;
  hence thesis by A2,A4,A5,Th2,INTEGRA5:13;
end;
