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
  A= [.0,1.] implies integral(sinh,A) = (number_e - 1)^2/(2*number_e)
proof
  exp_R(1)>0 by SIN_COS:55;
  then
A1: 2*number_e > 0 by IRRAT_1:def 7,XREAL_1:129;
  assume A=[.0,1.];
  then A=[.0,jj.];
  then upper_bound A=1 & lower_bound A=0 by Th37;
  then integral(sinh,A) = (number_e^2 + 1)/(2*number_e) - 1 by Th18,Th19,Th55
    .= (number_e^2 + 1)/(2*number_e) - (2*number_e)/(2*number_e) by A1,
XCMPLX_1:60
    .= ((number_e^2 + 1) - (2*number_e))/(2*number_e) by XCMPLX_1:120
    .= (number_e^2 - 2*number_e*1 + 1^2)/(2*number_e)
    .= (number_e - 1)^2/(2*number_e);
  hence thesis;
end;
