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