let A be non empty closed_interval Subset of REAL; :: thesis: integral (((AffineMap (1,0)) (#) cosh),A) = ((((AffineMap (1,0)) (#) sinh) - cosh) . (upper_bound A)) - ((((AffineMap (1,0)) (#) sinh) - cosh) . (lower_bound A))
A1: for x being Element of REAL st x in dom ((((AffineMap (1,0)) (#) sinh) - cosh) `| REAL) holds
((((AffineMap (1,0)) (#) sinh) - cosh) `| REAL) . x = ((AffineMap (1,0)) (#) cosh) . x

proof

let x be Element of REAL ; :: thesis: ( x in dom ((((AffineMap (1,0)) (#) sinh) - cosh) `| REAL) implies ((((AffineMap (1,0)) (#) sinh) - cosh) `| REAL) . x = ((AffineMap (1,0)) (#) cosh) . x )
assume x in dom ((((AffineMap (1,0)) (#) sinh) - cosh) `| REAL) ; :: thesis: ((((AffineMap (1,0)) (#) sinh) - cosh) `| REAL) . x = ((AffineMap (1,0)) (#) cosh) . x
((((AffineMap (1,0)) (#) sinh) - cosh) `| REAL) . x = ((1 * x) + 0) * (cosh . x) by Th11
.= ((AffineMap (1,0)) . x) * (cosh . x) by FCONT_1:def 4
.= ((AffineMap (1,0)) (#) cosh) . x by VALUED_1:5 ;
hence ((((AffineMap (1,0)) (#) sinh) - cosh) `| REAL) . x = ((AffineMap (1,0)) (#) cosh) . x ; :: thesis: verum

end;

A2: dom ((AffineMap (1,0)) (#) cosh) = [#] REAL by FUNCT_2:def 1;
then dom ((((AffineMap (1,0)) (#) sinh) - cosh) `| REAL) = dom ((AffineMap (1,0)) (#) cosh) by Th11, FDIFF_1:def 7;
then A3: (((AffineMap (1,0)) (#) sinh) - cosh) `| REAL = (AffineMap (1,0)) (#) cosh by A1, PARTFUN1:5;
( dom (AffineMap (1,0)) = [#] REAL & ( for x being Real 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)) (#) cosh) | REAL is continuous by A2, FDIFF_1:21, FDIFF_1:25, SIN_COS2:35;
then A5: ((AffineMap (1,0)) (#) cosh) | A is continuous by FCONT_1:16;
((AffineMap (1,0)) (#) cosh) | A is bounded by A2, A4, INTEGRA5:10;
hence integral (((AffineMap (1,0)) (#) cosh),A) = ((((AffineMap (1,0)) (#) sinh) - cosh) . (upper_bound A)) - ((((AffineMap (1,0)) (#) sinh) - cosh) . (lower_bound A)) by A2, A5, A3, Th11, INTEGRA5:11, INTEGRA5:13; :: thesis: verum