let A be closed-interval Subset of REAL ; :: thesis: integral ((AffineMap 1,0 ) (#) sinh ),A = ((((AffineMap 1,0 ) (#) cosh ) - sinh ) . (sup A)) - ((((AffineMap 1,0 ) (#) cosh ) - sinh ) . (inf A))
A1: ( dom (((AffineMap 1,0 ) (#) cosh ) - sinh ) = [#] REAL & dom ((AffineMap 1,0 ) (#) sinh ) = [#] REAL & dom (AffineMap 1,0 ) = [#] REAL ) by FUNCT_2:def 1;
for x being Real st x in REAL holds
(AffineMap 1,0 ) . x = (1 * x) + 0 by JORDAN16:def 3;
then AffineMap 1,0 is_differentiable_on REAL by A1, FDIFF_1:31;
then ((AffineMap 1,0 ) (#) sinh ) | REAL is continuous by A1, SIN_COS2:34, FDIFF_1:29, FDIFF_1:33;
then ((AffineMap 1,0 ) (#) sinh ) | A is continuous by FCONT_1:17;
then A2: ( (AffineMap 1,0 ) (#) sinh is_integrable_on A & ((AffineMap 1,0 ) (#) sinh ) | A is bounded ) by A1, INTEGRA5:10, INTEGRA5:11;
A3: 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;

dom ((((AffineMap 1,0 ) (#) cosh ) - sinh ) `| REAL ) = dom ((AffineMap 1,0 ) (#) sinh ) by A1, Th10, FDIFF_1:def 8;
then (((AffineMap 1,0 ) (#) cosh ) - sinh ) `| REAL = (AffineMap 1,0 ) (#) sinh by A3, PARTFUN1:34;
hence integral ((AffineMap 1,0 ) (#) sinh ),A = ((((AffineMap 1,0 ) (#) cosh ) - sinh ) . (sup A)) - ((((AffineMap 1,0 ) (#) cosh ) - sinh ) . (inf A)) by A2, Th10, INTEGRA5:13; :: thesis: verum