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 Th57:
  integral(cosh,A) = sinh.(upper_bound A)-sinh.(lower_bound A)
proof
A1: for x being Element of REAL st x in dom (sinh`|REAL) holds (sinh`|REAL).
  x = cosh.x
  proof
    let x be Element of REAL;
    assume x in dom (sinh`|REAL);
    (sinh`|REAL).x = diff(sinh,x) by FDIFF_1:def 7,SIN_COS2:34;
    hence thesis by SIN_COS2:34;
  end;
A2: cosh is_integrable_on A & cosh|A is bounded by Lm17;
  dom (sinh`|REAL) = dom cosh by FDIFF_1:def 7,SIN_COS2:30,34;
  then sinh`|REAL = cosh by A1,PARTFUN1:5;
  hence thesis by A2,INTEGRA5:13,SIN_COS2:34;
end;
