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+cosh,A) = number_e -1
proof
A1: number_e > 0 by IRRAT_1:def 7,SIN_COS:55;
  assume A = [.0,1.];
  then upper_bound A=1 & lower_bound A=0 by Th37;
  then integral(sinh+cosh,A) = cosh.1 - cosh.0 + sinh.1 - sinh.0 by Th76
    .= (number_e^2 + 1)/(2*number_e) + (number_e^2 - 1)/(2*number_e) - 1 by
Th17,Th18,Th19,SIN_COS2:16
    .= ((number_e^2 + 1) + (number_e^2 - 1))/(2*number_e) -1 by XCMPLX_1:62
    .= (2*number_e^2)/(2*number_e) -1
    .= number_e^2/number_e -1 by XCMPLX_1:91
    .= number_e*number_e/number_e -1
    .= number_e -1 by A1,XCMPLX_1:89;
  hence thesis;
end;
