let n be Element of NAT ; for A being non empty closed_interval Subset of REAL st n <> 0 holds
integral (((AffineMap (1,0)) (#) (cos * (AffineMap (n,0)))),A) = ((((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0))))) . (upper_bound A)) - ((((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0))))) . (lower_bound A))
let A be non empty closed_interval Subset of REAL; ( n <> 0 implies integral (((AffineMap (1,0)) (#) (cos * (AffineMap (n,0)))),A) = ((((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0))))) . (upper_bound A)) - ((((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0))))) . (lower_bound A)) )
assume A1:
n <> 0
; integral (((AffineMap (1,0)) (#) (cos * (AffineMap (n,0)))),A) = ((((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0))))) . (upper_bound A)) - ((((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (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:
for x being Real st x in REAL holds
(AffineMap (1,0)) . x = x
A5:
for x being Element of REAL st x in dom ((((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0))))) `| REAL) holds
((((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0))))) `| REAL) . x = ((AffineMap (1,0)) (#) (cos * (AffineMap (n,0)))) . x
proof
let x be
Element of
REAL ;
( x in dom ((((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0))))) `| REAL) implies ((((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0))))) `| REAL) . x = ((AffineMap (1,0)) (#) (cos * (AffineMap (n,0)))) . x )
assume
x in dom ((((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0))))) `| REAL)
;
((((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0))))) `| REAL) . x = ((AffineMap (1,0)) (#) (cos * (AffineMap (n,0)))) . x
((((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0))))) `| REAL) . x =
x * (cos . (n * x))
by A1, Th9
.=
x * (cos . ((AffineMap (n,0)) . x))
by A2
.=
x * ((cos * (AffineMap (n,0))) . x)
by A3, FUNCT_1:12
.=
((AffineMap (1,0)) . x) * ((cos * (AffineMap (n,0))) . x)
by A4
.=
((AffineMap (1,0)) (#) (cos * (AffineMap (n,0)))) . x
by VALUED_1:5
;
hence
((((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0))))) `| REAL) . x = ((AffineMap (1,0)) (#) (cos * (AffineMap (n,0)))) . x
;
verum
end;
A6:
dom ((AffineMap (1,0)) (#) (cos * (AffineMap (n,0)))) = [#] REAL
by FUNCT_2:def 1;
((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0)))) is_differentiable_on REAL
by A1, Th9;
then
dom ((((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0))))) `| REAL) = dom ((AffineMap (1,0)) (#) (cos * (AffineMap (n,0))))
by A6, FDIFF_1:def 7;
then A7:
(((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0))))) `| REAL = (AffineMap (1,0)) (#) (cos * (AffineMap (n,0)))
by A5, PARTFUN1:5;
((AffineMap (1,0)) (#) (cos * (AffineMap (n,0)))) | A is continuous
;
then A8:
(AffineMap (1,0)) (#) (cos * (AffineMap (n,0))) is_integrable_on A
by A6, INTEGRA5:11;
((AffineMap (1,0)) (#) (cos * (AffineMap (n,0)))) | A is bounded
by A6, INTEGRA5:10;
hence
integral (((AffineMap (1,0)) (#) (cos * (AffineMap (n,0)))),A) = ((((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0))))) . (upper_bound A)) - ((((1 / (n ^2)) (#) (cos * (AffineMap (n,0)))) + ((AffineMap ((1 / n),0)) (#) (sin * (AffineMap (n,0))))) . (lower_bound A))
by A1, A8, A7, Th9, INTEGRA5:13; verum