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 Th55:
  integral(sinh,A) = cosh.(upper_bound A)-cosh.(lower_bound A)
proof
A1: for x being Element of REAL st x in dom (cosh`|REAL) holds (cosh`|REAL).
  x = sinh.x
  proof
    let x be Element of REAL;
    assume x in dom (cosh`|REAL);
    (cosh`|REAL).x = diff(cosh,x) by FDIFF_1:def 7,SIN_COS2:35;
    hence thesis by SIN_COS2:35;
  end;
A2: sinh is_integrable_on A & sinh|A is bounded by Lm15;
  dom (cosh`|REAL) = dom sinh by FDIFF_1:def 7,SIN_COS2:30,35;
  then cosh`|REAL = sinh by A1,PARTFUN1:5;
  hence thesis by A2,INTEGRA5:13,SIN_COS2:35;
end;
