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 Th76:
  integral(sinh+cosh,A) = cosh.(upper_bound A) - cosh.(lower_bound
  A) + sinh.(upper_bound A) - sinh.(lower_bound A)
proof
A1: [#]REAL is open Subset of REAL;
  cosh|A is continuous by Lm16;
  then
A2: cosh is_integrable_on A by Lm10,INTEGRA5:11;
  sinh|A is continuous by Lm14;
  then
A3: sinh is_integrable_on A by Lm9,INTEGRA5:11;
  cosh|A is bounded & sinh|A is bounded by Lm9,Lm10,Lm14,Lm16,INTEGRA5:10;
  hence thesis by A2,A3,A1,Th30,Th31,Th66,SIN_COS2:34,35;
end;
