let a, b be Real; for Z being open Subset of REAL
for f being PartFunc of REAL,REAL st Z c= dom ((- (2 / (3 * b))) (#) ((#R (3 / 2)) * f)) & ( for x being Real st x in Z holds
( f . x = a - (b * x) & b <> 0 & f . x > 0 ) ) holds
( (- (2 / (3 * b))) (#) ((#R (3 / 2)) * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((- (2 / (3 * b))) (#) ((#R (3 / 2)) * f)) `| Z) . x = (a - (b * x)) #R (1 / 2) ) )
let Z be open Subset of REAL; for f being PartFunc of REAL,REAL st Z c= dom ((- (2 / (3 * b))) (#) ((#R (3 / 2)) * f)) & ( for x being Real st x in Z holds
( f . x = a - (b * x) & b <> 0 & f . x > 0 ) ) holds
( (- (2 / (3 * b))) (#) ((#R (3 / 2)) * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((- (2 / (3 * b))) (#) ((#R (3 / 2)) * f)) `| Z) . x = (a - (b * x)) #R (1 / 2) ) )
let f be PartFunc of REAL,REAL; ( Z c= dom ((- (2 / (3 * b))) (#) ((#R (3 / 2)) * f)) & ( for x being Real st x in Z holds
( f . x = a - (b * x) & b <> 0 & f . x > 0 ) ) implies ( (- (2 / (3 * b))) (#) ((#R (3 / 2)) * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((- (2 / (3 * b))) (#) ((#R (3 / 2)) * f)) `| Z) . x = (a - (b * x)) #R (1 / 2) ) ) )
assume that
A1:
Z c= dom ((- (2 / (3 * b))) (#) ((#R (3 / 2)) * f))
and
A2:
for x being Real st x in Z holds
( f . x = a - (b * x) & b <> 0 & f . x > 0 )
; ( (- (2 / (3 * b))) (#) ((#R (3 / 2)) * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((- (2 / (3 * b))) (#) ((#R (3 / 2)) * f)) `| Z) . x = (a - (b * x)) #R (1 / 2) ) )
A3:
Z c= dom ((#R (3 / 2)) * f)
by A1, VALUED_1:def 5;
then
for y being object st y in Z holds
y in dom f
by FUNCT_1:11;
then A4:
Z c= dom f
by TARSKI:def 3;
A5:
for x being Real st x in Z holds
f . x = ((- b) * x) + a
proof
let x be
Real;
( x in Z implies f . x = ((- b) * x) + a )
assume
x in Z
;
f . x = ((- b) * x) + a
then
f . x = a - (b * x)
by A2;
hence
f . x = ((- b) * x) + a
;
verum
end;
then A6:
f is_differentiable_on Z
by A4, FDIFF_1:23;
then A7:
(#R (3 / 2)) * f is_differentiable_on Z
by A3, FDIFF_1:9;
for x being Real st x in Z holds
(((- (2 / (3 * b))) (#) ((#R (3 / 2)) * f)) `| Z) . x = (a - (b * x)) #R (1 / 2)
proof
let x be
Real;
( x in Z implies (((- (2 / (3 * b))) (#) ((#R (3 / 2)) * f)) `| Z) . x = (a - (b * x)) #R (1 / 2) )
assume A8:
x in Z
;
(((- (2 / (3 * b))) (#) ((#R (3 / 2)) * f)) `| Z) . x = (a - (b * x)) #R (1 / 2)
then A9:
3
* b <> 0
by A2;
A10:
f . x = a - (b * x)
by A2, A8;
A11:
(
f is_differentiable_in x &
f . x > 0 )
by A2, A6, A8, FDIFF_1:9;
(((- (2 / (3 * b))) (#) ((#R (3 / 2)) * f)) `| Z) . x =
(- (2 / (3 * b))) * (diff (((#R (3 / 2)) * f),x))
by A1, A7, A8, FDIFF_1:20
.=
(- (2 / (3 * b))) * (((3 / 2) * ((f . x) #R ((3 / 2) - 1))) * (diff (f,x)))
by A11, TAYLOR_1:22
.=
(- (2 / (3 * b))) * (((3 / 2) * ((f . x) #R ((3 / 2) - 1))) * ((f `| Z) . x))
by A6, A8, FDIFF_1:def 7
.=
(- (2 / (3 * b))) * (((3 / 2) * ((a - (b * x)) #R ((3 / 2) - 1))) * (- b))
by A4, A5, A8, A10, FDIFF_1:23
.=
((2 / (3 * b)) * ((3 * b) / 2)) * ((a - (b * x)) #R ((3 / 2) - 1))
.=
1
* ((a - (b * x)) #R ((3 / 2) - 1))
by A9, XCMPLX_1:112
.=
(a - (b * x)) #R (1 / 2)
;
hence
(((- (2 / (3 * b))) (#) ((#R (3 / 2)) * f)) `| Z) . x = (a - (b * x)) #R (1 / 2)
;
verum
end;
hence
( (- (2 / (3 * b))) (#) ((#R (3 / 2)) * f) is_differentiable_on Z & ( for x being Real st x in Z holds
(((- (2 / (3 * b))) (#) ((#R (3 / 2)) * f)) `| Z) . x = (a - (b * x)) #R (1 / 2) ) )
by A1, A7, FDIFF_1:20; verum