let n, m be Element of NAT ; for A being non empty closed_interval Subset of REAL st m + n <> 0 & m - n <> 0 holds
integral (((sin * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0)))),A) = (((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) . (upper_bound A)) - (((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) . (lower_bound A))
let A be non empty closed_interval Subset of REAL; ( m + n <> 0 & m - n <> 0 implies integral (((sin * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0)))),A) = (((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) . (upper_bound A)) - (((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) . (lower_bound A)) )
assume A1:
( m + n <> 0 & m - n <> 0 )
; integral (((sin * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0)))),A) = (((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) . (upper_bound A)) - (((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) . (lower_bound A))
A2:
for x being Real st x in REAL holds
(AffineMap (n,0)) . x = n * x
A3:
dom (cos * (AffineMap (n,0))) = [#] REAL
by FUNCT_2:def 1;
A4:
dom (sin * (AffineMap (m,0))) = [#] REAL
by FUNCT_2:def 1;
A5:
for x being Real st x in REAL holds
(AffineMap (m,0)) . x = m * x
A6:
for x being Element of REAL st x in dom (((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) `| REAL) holds
(((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) `| REAL) . x = ((sin * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0)))) . x
proof
let x be
Element of
REAL ;
( x in dom (((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) `| REAL) implies (((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) `| REAL) . x = ((sin * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0)))) . x )
assume
x in dom (((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) `| REAL)
;
(((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) `| REAL) . x = ((sin * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0)))) . x
(((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) `| REAL) . x =
(sin . (m * x)) * (cos . (n * x))
by A1, Th7
.=
(sin . ((AffineMap (m,0)) . x)) * (cos . (n * x))
by A5
.=
(sin . ((AffineMap (m,0)) . x)) * (cos . ((AffineMap (n,0)) . x))
by A2
.=
((sin * (AffineMap (m,0))) . x) * (cos . ((AffineMap (n,0)) . x))
by A4, FUNCT_1:12
.=
((sin * (AffineMap (m,0))) . x) * ((cos * (AffineMap (n,0))) . x)
by A3, FUNCT_1:12
.=
((sin * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0)))) . x
by VALUED_1:5
;
hence
(((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) `| REAL) . x = ((sin * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0)))) . x
;
verum
end;
A7:
[#] REAL = dom ((sin * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0))))
by FUNCT_2:def 1;
((sin * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0)))) | A is continuous
;
then A8:
(sin * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0))) is_integrable_on A
by A7, INTEGRA5:11;
(- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0)))) is_differentiable_on REAL
by A1, Th7;
then
dom (((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) `| REAL) = dom ((sin * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0))))
by A7, FDIFF_1:def 7;
then A9:
((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) `| REAL = (sin * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0)))
by A6, PARTFUN1:5;
((sin * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0)))) | A is bounded
by A7, INTEGRA5:10;
hence
integral (((sin * (AffineMap (m,0))) (#) (cos * (AffineMap (n,0)))),A) = (((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) . (upper_bound A)) - (((- ((1 / (2 * (m + n))) (#) (cos * (AffineMap ((m + n),0))))) - ((1 / (2 * (m - n))) (#) (cos * (AffineMap ((m - n),0))))) . (lower_bound A))
by A1, A8, A9, Th7, INTEGRA5:13; verum