let A be non empty closed_interval Subset of REAL; for f, g, f1, f2 being PartFunc of REAL,REAL
for Z being open Subset of REAL st A c= Z & f1 = g - f2 & f2 = #Z 2 & ( for x being Real st x in Z holds
( f . x = x * ((1 - (x #Z 2)) #R (- (1 / 2))) & g . x = 1 & f1 . x > 0 ) ) & Z c= dom ((#R (1 / 2)) * f1) & Z = dom f & f | A is continuous holds
integral (f,A) = ((- ((#R (1 / 2)) * f1)) . (upper_bound A)) - ((- ((#R (1 / 2)) * f1)) . (lower_bound A))
let f, g, f1, f2 be PartFunc of REAL,REAL; for Z being open Subset of REAL st A c= Z & f1 = g - f2 & f2 = #Z 2 & ( for x being Real st x in Z holds
( f . x = x * ((1 - (x #Z 2)) #R (- (1 / 2))) & g . x = 1 & f1 . x > 0 ) ) & Z c= dom ((#R (1 / 2)) * f1) & Z = dom f & f | A is continuous holds
integral (f,A) = ((- ((#R (1 / 2)) * f1)) . (upper_bound A)) - ((- ((#R (1 / 2)) * f1)) . (lower_bound A))
let Z be open Subset of REAL; ( A c= Z & f1 = g - f2 & f2 = #Z 2 & ( for x being Real st x in Z holds
( f . x = x * ((1 - (x #Z 2)) #R (- (1 / 2))) & g . x = 1 & f1 . x > 0 ) ) & Z c= dom ((#R (1 / 2)) * f1) & Z = dom f & f | A is continuous implies integral (f,A) = ((- ((#R (1 / 2)) * f1)) . (upper_bound A)) - ((- ((#R (1 / 2)) * f1)) . (lower_bound A)) )
assume A1:
( A c= Z & f1 = g - f2 & f2 = #Z 2 & ( for x being Real st x in Z holds
( f . x = x * ((1 - (x #Z 2)) #R (- (1 / 2))) & g . x = 1 & f1 . x > 0 ) ) & Z c= dom ((#R (1 / 2)) * f1) & Z = dom f & f | A is continuous )
; integral (f,A) = ((- ((#R (1 / 2)) * f1)) . (upper_bound A)) - ((- ((#R (1 / 2)) * f1)) . (lower_bound A))
then A2:
( f is_integrable_on A & f | A is bounded )
by INTEGRA5:10, INTEGRA5:11;
A3:
for x being Real st x in Z holds
( g . x = 1 & f1 . x > 0 )
by A1;
A4:
Z c= dom (- ((#R (1 / 2)) * f1))
by A1, VALUED_1:8;
for y being object st y in Z holds
y in dom f1
by A1, FUNCT_1:11;
then A5:
Z c= dom (g + ((- 1) (#) f2))
by A1;
A6:
(#R (1 / 2)) * f1 is_differentiable_on Z
by A1, A3, FDIFF_7:22;
then A7:
(- 1) (#) ((#R (1 / 2)) * f1) is_differentiable_on Z
by A4, FDIFF_1:20;
A8:
( f2 = #Z 2 & ( for x being Real st x in Z holds
g . x = 1 + (0 * x) ) )
by A1;
then A9:
( f1 is_differentiable_on Z & ( for x being Real st x in Z holds
(f1 `| Z) . x = 0 + ((2 * (- 1)) * x) ) )
by A1, A5, FDIFF_4:12;
A10:
for x being Real st x in Z holds
((- ((#R (1 / 2)) * f1)) `| Z) . x = x * ((1 - (x #Z 2)) #R (- (1 / 2)))
proof
let x be
Real;
( x in Z implies ((- ((#R (1 / 2)) * f1)) `| Z) . x = x * ((1 - (x #Z 2)) #R (- (1 / 2))) )
assume A11:
x in Z
;
((- ((#R (1 / 2)) * f1)) `| Z) . x = x * ((1 - (x #Z 2)) #R (- (1 / 2)))
then A12:
x in dom (g - f2)
by A1, FUNCT_1:11;
A13:
f1 is_differentiable_in x
by A9, A11, FDIFF_1:9;
A14:
(g - f2) . x =
(g . x) - (f2 . x)
by A12, VALUED_1:13
.=
1
- (f2 . x)
by A1, A11
.=
1
- (x #Z 2)
by A1, TAYLOR_1:def 1
;
then A15:
(
f1 . x = 1
- (x #Z 2) &
f1 . x > 0 )
by A1, A11;
A16:
(#R (1 / 2)) * f1 is_differentiable_in x
by A6, A11, FDIFF_1:9;
((- ((#R (1 / 2)) * f1)) `| Z) . x =
diff (
(- ((#R (1 / 2)) * f1)),
x)
by A7, A11, FDIFF_1:def 7
.=
(- 1) * (diff (((#R (1 / 2)) * f1),x))
by A16, FDIFF_1:15
.=
(- 1) * (((1 / 2) * ((f1 . x) #R ((1 / 2) - 1))) * (diff (f1,x)))
by A13, A15, TAYLOR_1:22
.=
(- 1) * (((1 / 2) * ((f1 . x) #R ((1 / 2) - 1))) * ((f1 `| Z) . x))
by A9, A11, FDIFF_1:def 7
.=
(- 1) * (((1 / 2) * ((f1 . x) #R ((1 / 2) - 1))) * (0 + ((2 * (- 1)) * x)))
by A1, A5, A8, A11, FDIFF_4:12
.=
x * ((1 - (x #Z 2)) #R (- (1 / 2)))
by A1, A14
;
hence
((- ((#R (1 / 2)) * f1)) `| Z) . x = x * ((1 - (x #Z 2)) #R (- (1 / 2)))
;
verum
end;
A17:
for x being Element of REAL st x in dom ((- ((#R (1 / 2)) * f1)) `| Z) holds
((- ((#R (1 / 2)) * f1)) `| Z) . x = f . x
dom ((- ((#R (1 / 2)) * f1)) `| Z) = dom f
by A1, A7, FDIFF_1:def 7;
then
(- ((#R (1 / 2)) * f1)) `| Z = f
by A17, PARTFUN1:5;
hence
integral (f,A) = ((- ((#R (1 / 2)) * f1)) . (upper_bound A)) - ((- ((#R (1 / 2)) * f1)) . (lower_bound A))
by A1, A2, A7, INTEGRA5:13; verum