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
  integral(sinh(#)cosh,A) =1/2*((cosh.(upper_bound A))*cosh.(upper_bound
  A)- (cosh).(lower_bound A)*cosh.(lower_bound A))
proof
  sinh|A is continuous by Lm14;
  then
A1: (cosh)`|REAL is_integrable_on A by Lm9,Th31,INTEGRA5:11;
  (cosh`|REAL)|A is bounded & [#]REAL is open Subset of REAL by Lm9,Lm14,Th31,
INTEGRA5:10;
  then
  integral(sinh(#)cosh,A) =(cosh.(upper_bound A))*cosh.(upper_bound A)- (
  cosh).(lower_bound A)*cosh.(lower_bound A) -integral(sinh(#)cosh,A) by A1
,Th31,INTEGRA5:21,SIN_COS2:35;
  hence thesis;
end;
