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
  integral((AffineMap(1,0))(#)exp_R,A) = (exp_R(#)(AffineMap(1,-1))).(
  upper_bound A) -(exp_R(#)(AffineMap(1,-1))).(lower_bound A)
proof
A1: for x st x in REAL holds (AffineMap(1,-1)).x = x-1
  proof
    let x;
    assume x in REAL;
    (AffineMap(1,-1)).x = 1*x + (-1) by FCONT_1:def 4
      .= x-1;
    hence thesis;
  end;
A2: dom (exp_R(#)(AffineMap(1,-1))) = [#]REAL by FUNCT_2:def 1;
A3: for x st x in REAL holds ((AffineMap(1,0))(#)exp_R).x = x*(exp_R.x)
  proof
    let x;
    assume x in REAL;
    ((AffineMap(1,0))(#)exp_R).x = ((AffineMap(1,0)).x)*(exp_R.x) by VALUED_1:5
      .= (1*x+0)*(exp_R.x) by FCONT_1:def 4
      .= x*(exp_R.x);
    hence thesis;
  end;
A4: for x being Element of REAL
st x in dom ((exp_R(#)AffineMap(1,-1))`|REAL) holds ((exp_R(#)
  AffineMap(1,-1))`|REAL).x = ((AffineMap(1,0))(#)exp_R).x
  proof
    let x be Element of REAL;
    assume
x in dom ((exp_R(#)AffineMap(1,-1))`|REAL);
    ((exp_R(#)AffineMap(1,-1))`|REAL).x = x*(exp_R.x) by A2,A1,FDIFF_4:55
      .= ((AffineMap(1,0))(#)exp_R).x by A3;
    hence thesis;
  end;
A5: [#]REAL = dom ((AffineMap(1,0))(#)exp_R) by FUNCT_2:def 1;
  ((AffineMap(1,0))(#)exp_R)|A is continuous;
  then
A6: (AffineMap(1,0))(#)exp_R is_integrable_on A by A5,INTEGRA5:11;
  (exp_R(#)(AffineMap(1,-1))) is_differentiable_on REAL by A2,A1,FDIFF_4:55;
  then dom ((exp_R(#)AffineMap(1,-1))`|REAL) = dom ((AffineMap(1,0))(#)exp_R)
  by A5,FDIFF_1:def 7;
  then
A7: ((exp_R(#)AffineMap(1,-1))`|REAL) = (AffineMap(1,0))(#)exp_R by A4,
PARTFUN1:5;
  ((AffineMap(1,0))(#)exp_R)|A is bounded by A5,INTEGRA5:10;
  hence thesis by A2,A1,A6,A7,FDIFF_4:55,INTEGRA5:13;
end;
