let A be closed-interval Subset of REAL; :: thesis: integral (((AffineMap (1,0)) (#) sinh),A) = ((((AffineMap (1,0)) (#) cosh) - sinh) . (upper_bound A)) - ((((AffineMap (1,0)) (#) cosh) - sinh) . (lower_bound A))
A1: for x being 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 Real; :: thesis: ( x in dom ((((AffineMap (1,0)) (#) cosh) - sinh) `| REAL) implies ((((AffineMap (1,0)) (#) cosh) - sinh) `| REAL) . x = ((AffineMap (1,0)) (#) sinh) . x )
assume x in dom ((((AffineMap (1,0)) (#) cosh) - sinh) `| REAL) ; :: thesis: ((((AffineMap (1,0)) (#) cosh) - sinh) `| REAL) . x = ((AffineMap (1,0)) (#) sinh) . x
((((AffineMap (1,0)) (#) cosh) - sinh) `| REAL) . x = ((1 * x) + 0) * (sinh . x) by Th10
.= ((AffineMap (1,0)) . x) * (sinh . x) by JORDAN16:def 3
.= ((AffineMap (1,0)) (#) sinh) . x by VALUED_1:5 ;
hence ((((AffineMap (1,0)) (#) cosh) - sinh) `| REAL) . x = ((AffineMap (1,0)) (#) sinh) . x ; :: thesis: verum

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 8;
then A3: (((AffineMap (1,0)) (#) cosh) - sinh) `| REAL = (AffineMap (1,0)) (#) sinh by A1, PARTFUN1:34;
( dom (AffineMap (1,0)) = [#] REAL & ( for x being Real st x in REAL holds
(AffineMap (1,0)) . x = (1 * x) + 0 ) ) by FUNCT_2:def 1, JORDAN16:def 3;
then AffineMap (1,0) is_differentiable_on REAL by FDIFF_1:31;
then A4: ((AffineMap (1,0)) (#) sinh) | REAL is continuous by A2, FDIFF_1:29, FDIFF_1:33, SIN_COS2:34;
then A5: ((AffineMap (1,0)) (#) sinh) | A is continuous by FCONT_1:17;
((AffineMap (1,0)) (#) sinh) | A is bounded by A2, A4, FCONT_1:17, INTEGRA5:10;
hence integral (((AffineMap (1,0)) (#) sinh),A) = ((((AffineMap (1,0)) (#) cosh) - sinh) . (upper_bound A)) - ((((AffineMap (1,0)) (#) cosh) - sinh) . (lower_bound A)) by A2, A5, A3, Th10, INTEGRA5:11, INTEGRA5:13; :: thesis: verum