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 Th53:
  integral(exp_R,A) = exp_R.(upper_bound A)-exp_R.(lower_bound A)
proof
A1: for x being Element of REAL st x in dom (exp_R`|REAL) holds (exp_R`|REAL
  ).x = exp_R.x
  proof
    let x be Element of REAL;
    assume x in dom (exp_R`|REAL);
    (exp_R`|REAL).x = diff(exp_R,x) by FDIFF_1:def 7,SIN_COS:66;
    hence thesis by SIN_COS:65;
  end;
A2: exp_R is_integrable_on A & (exp_R)|A is bounded by Lm13;
  dom (exp_R`|REAL) = dom exp_R by FDIFF_1:def 7,SIN_COS:47,66;
  then exp_R`|REAL = exp_R by A1,PARTFUN1:5;
  hence thesis by A2,INTEGRA5:13,SIN_COS:66;
end;
