reserve r,x,x0,a,b for Real;
reserve n,m for Element of NAT;
reserve A for non empty closed_interval Subset of REAL;
reserve Z for open Subset of REAL;

theorem
  integral((AffineMap(1,0))(#)sinh,A) = ((AffineMap(1,0))(#)cosh-sinh).(
  upper_bound A) -((AffineMap(1,0))(#)cosh-sinh).(lower_bound A)
proof
A1: for x being Element of REAL
st x in dom (((AffineMap(1,0))(#)cosh-sinh)`|REAL) holds (((
  AffineMap(1,0))(#)cosh-sinh)`|REAL).x = ((AffineMap(1,0))(#)sinh).x
  proof
    let x be Element of REAL;
    assume x in dom (((AffineMap(1,0))(#)cosh-sinh)`|REAL);
    (((AffineMap(1,0))(#)cosh-sinh)`|REAL).x = (1*x+0)*sinh.x by Th10
      .= ((AffineMap(1,0)).x)*sinh.x by FCONT_1:def 4
      .= ((AffineMap(1,0))(#)sinh).x by VALUED_1:5;
    hence thesis;
  end;
A2: dom ((AffineMap(1,0))(#)sinh) = [#]REAL by FUNCT_2:def 1;
  then dom (((AffineMap(1,0))(#)cosh-sinh)`|REAL) = dom ((AffineMap(1,0))(#)
  sinh) by Th10,FDIFF_1:def 7;
  then
A3: (((AffineMap(1,0))(#)cosh-sinh)`|REAL) = (AffineMap(1,0))(#)sinh by A1,
PARTFUN1:5;
  dom (AffineMap(1,0)) = [#]REAL & for x st x in REAL holds AffineMap(1,0)
  .x=1 *x + 0 by FCONT_1:def 4,FUNCT_2:def 1;
  then AffineMap(1,0) is_differentiable_on REAL by FDIFF_1:23;
  then
A4: ((AffineMap(1,0))(#)sinh)|REAL is continuous by A2,FDIFF_1:21,25
,SIN_COS2:34;
  then
A5: ((AffineMap(1,0))(#)sinh)|A is continuous by FCONT_1:16;
  ((AffineMap(1,0))(#)sinh)|A is bounded by A2,A4,INTEGRA5:10;
  hence thesis by A2,A5,A3,Th10,INTEGRA5:11,13;
end;
