PDIFF_3: Second-order Partial Differentiation of Real Binary Functions

:: Second-order Partial Differentiation of Real Binary Functions
:: by Bing Xie , Xiquan Liang and Xiuzhuan Shen
::
:: Received December 16, 2008
:: Copyright (c) 2008-2021 Association of Mizar Users

registration

cluster Function-like RestFunc-like -> total for Element of K16(K17(REAL,REAL));

coherence by FDIFF_1:def 2;

end;

definition

let i be Element of NAT ;
let n be non zero Element of NAT ;
let f be PartFunc of (REAL n),REAL;

func pdiff1 (f,i) -> Function of (REAL n),REAL means :: PDIFF_3:def 1
for z being Element of REAL n holds it . z = partdiff (f,z,i);

existence

proof end;

uniqueness

proof end;

end;

:: deftheorem defines pdiff1 PDIFF_3:def 1 :

definition

let f be PartFunc of (REAL 2),REAL;
let z be Element of REAL 2;

pred f is_hpartial_differentiable`11_in z means :: PDIFF_3:def 2
ex x0, y0 being Real st
( z = <*x0,y0*> & ex N being Neighbourhood of x0 st
( N c= dom (SVF1 (1,(pdiff1 (f,1)),z)) & ex L being LinearFunc ex R being RestFunc st
for x being Real st x in N holds
((SVF1 (1,(pdiff1 (f,1)),z)) . x) - ((SVF1 (1,(pdiff1 (f,1)),z)) . x0) = (L . (x - x0)) + (R . (x - x0)) ) );

pred f is_hpartial_differentiable`12_in z means :: PDIFF_3:def 3
ex x0, y0 being Real st
( z = <*x0,y0*> & ex N being Neighbourhood of y0 st
( N c= dom (SVF1 (2,(pdiff1 (f,1)),z)) & ex L being LinearFunc ex R being RestFunc st
for y being Real st y in N holds
((SVF1 (2,(pdiff1 (f,1)),z)) . y) - ((SVF1 (2,(pdiff1 (f,1)),z)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) );

pred f is_hpartial_differentiable`21_in z means :: PDIFF_3:def 4
ex x0, y0 being Real st
( z = <*x0,y0*> & ex N being Neighbourhood of x0 st
( N c= dom (SVF1 (1,(pdiff1 (f,2)),z)) & ex L being LinearFunc ex R being RestFunc st
for x being Real st x in N holds
((SVF1 (1,(pdiff1 (f,2)),z)) . x) - ((SVF1 (1,(pdiff1 (f,2)),z)) . x0) = (L . (x - x0)) + (R . (x - x0)) ) );

pred f is_hpartial_differentiable`22_in z means :: PDIFF_3:def 5
ex x0, y0 being Real st
( z = <*x0,y0*> & ex N being Neighbourhood of y0 st
( N c= dom (SVF1 (2,(pdiff1 (f,2)),z)) & ex L being LinearFunc ex R being RestFunc st
for y being Real st y in N holds
((SVF1 (2,(pdiff1 (f,2)),z)) . y) - ((SVF1 (2,(pdiff1 (f,2)),z)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) );

end;

:: deftheorem defines is_hpartial_differentiable`11_in PDIFF_3:def 2 :

:: deftheorem defines is_hpartial_differentiable`12_in PDIFF_3:def 3 :

:: deftheorem defines is_hpartial_differentiable`21_in PDIFF_3:def 4 :

:: deftheorem defines is_hpartial_differentiable`22_in PDIFF_3:def 5 :

definition

let f be PartFunc of (REAL 2),REAL;
let z be Element of REAL 2;
assume A1: f is_hpartial_differentiable`11_in z ;

func hpartdiff11 (f,z) -> Real means :Def6: :: PDIFF_3:def 6
ex x0, y0 being Real st
( z = <*x0,y0*> & ex N being Neighbourhood of x0 st
( N c= dom (SVF1 (1,(pdiff1 (f,1)),z)) & ex L being LinearFunc ex R being RestFunc st
( it = L . 1 & ( for x being Real st x in N holds
((SVF1 (1,(pdiff1 (f,1)),z)) . x) - ((SVF1 (1,(pdiff1 (f,1)),z)) . x0) = (L . (x - x0)) + (R . (x - x0)) ) ) ) );

existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def6 defines hpartdiff11 PDIFF_3:def 6 :

definition

let f be PartFunc of (REAL 2),REAL;
let z be Element of REAL 2;
assume A1: f is_hpartial_differentiable`12_in z ;

func hpartdiff12 (f,z) -> Real means :Def7: :: PDIFF_3:def 7
ex x0, y0 being Real st
( z = <*x0,y0*> & ex N being Neighbourhood of y0 st
( N c= dom (SVF1 (2,(pdiff1 (f,1)),z)) & ex L being LinearFunc ex R being RestFunc st
( it = L . 1 & ( for y being Real st y in N holds
((SVF1 (2,(pdiff1 (f,1)),z)) . y) - ((SVF1 (2,(pdiff1 (f,1)),z)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) ) );

existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def7 defines hpartdiff12 PDIFF_3:def 7 :

definition

let f be PartFunc of (REAL 2),REAL;
let z be Element of REAL 2;
assume A1: f is_hpartial_differentiable`21_in z ;

func hpartdiff21 (f,z) -> Real means :Def8: :: PDIFF_3:def 8
ex x0, y0 being Real st
( z = <*x0,y0*> & ex N being Neighbourhood of x0 st
( N c= dom (SVF1 (1,(pdiff1 (f,2)),z)) & ex L being LinearFunc ex R being RestFunc st
( it = L . 1 & ( for x being Real st x in N holds
((SVF1 (1,(pdiff1 (f,2)),z)) . x) - ((SVF1 (1,(pdiff1 (f,2)),z)) . x0) = (L . (x - x0)) + (R . (x - x0)) ) ) ) );

existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def8 defines hpartdiff21 PDIFF_3:def 8 :

definition

let f be PartFunc of (REAL 2),REAL;
let z be Element of REAL 2;
assume A1: f is_hpartial_differentiable`22_in z ;

func hpartdiff22 (f,z) -> Real means :Def9: :: PDIFF_3:def 9
ex x0, y0 being Real st
( z = <*x0,y0*> & ex N being Neighbourhood of y0 st
( N c= dom (SVF1 (2,(pdiff1 (f,2)),z)) & ex L being LinearFunc ex R being RestFunc st
( it = L . 1 & ( for y being Real st y in N holds
((SVF1 (2,(pdiff1 (f,2)),z)) . y) - ((SVF1 (2,(pdiff1 (f,2)),z)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) ) );

existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def9 defines hpartdiff22 PDIFF_3:def 9 :

theorem :: PDIFF_3:1

for x0, y0 being Real
for z being Element of REAL 2
for f being PartFunc of (REAL 2),REAL st z = <*x0,y0*> & f is_hpartial_differentiable`11_in z holds
SVF1 (1,(pdiff1 (f,1)),z) is_differentiable_in x0

proof end;

theorem :: PDIFF_3:2

for x0, y0 being Real
for z being Element of REAL 2
for f being PartFunc of (REAL 2),REAL st z = <*x0,y0*> & f is_hpartial_differentiable`12_in z holds
SVF1 (2,(pdiff1 (f,1)),z) is_differentiable_in y0

proof end;

theorem :: PDIFF_3:3

for x0, y0 being Real
for z being Element of REAL 2
for f being PartFunc of (REAL 2),REAL st z = <*x0,y0*> & f is_hpartial_differentiable`21_in z holds
SVF1 (1,(pdiff1 (f,2)),z) is_differentiable_in x0

proof end;

theorem :: PDIFF_3:4

for x0, y0 being Real
for z being Element of REAL 2
for f being PartFunc of (REAL 2),REAL st z = <*x0,y0*> & f is_hpartial_differentiable`22_in z holds
SVF1 (2,(pdiff1 (f,2)),z) is_differentiable_in y0

proof end;

theorem Th5: :: PDIFF_3:5

for x0, y0 being Real
for z being Element of REAL 2
for f being PartFunc of (REAL 2),REAL st z = <*x0,y0*> & f is_hpartial_differentiable`11_in z holds
hpartdiff11 (f,z) = diff ((SVF1 (1,(pdiff1 (f,1)),z)),x0)

proof end;

theorem Th6: :: PDIFF_3:6

for x0, y0 being Real
for z being Element of REAL 2
for f being PartFunc of (REAL 2),REAL st z = <*x0,y0*> & f is_hpartial_differentiable`12_in z holds
hpartdiff12 (f,z) = diff ((SVF1 (2,(pdiff1 (f,1)),z)),y0)

proof end;

theorem Th7: :: PDIFF_3:7

for x0, y0 being Real
for z being Element of REAL 2
for f being PartFunc of (REAL 2),REAL st z = <*x0,y0*> & f is_hpartial_differentiable`21_in z holds
hpartdiff21 (f,z) = diff ((SVF1 (1,(pdiff1 (f,2)),z)),x0)

proof end;

theorem Th8: :: PDIFF_3:8

for x0, y0 being Real
for z being Element of REAL 2
for f being PartFunc of (REAL 2),REAL st z = <*x0,y0*> & f is_hpartial_differentiable`22_in z holds
hpartdiff22 (f,z) = diff ((SVF1 (2,(pdiff1 (f,2)),z)),y0)

proof end;

definition

let f be PartFunc of (REAL 2),REAL;
let Z be set ;

pred f is_hpartial_differentiable`11_on Z means :: PDIFF_3:def 10
( Z c= dom f & ( for z being Element of REAL 2 st z in Z holds
f | Z is_hpartial_differentiable`11_in z ) );

pred f is_hpartial_differentiable`12_on Z means :: PDIFF_3:def 11
( Z c= dom f & ( for z being Element of REAL 2 st z in Z holds
f | Z is_hpartial_differentiable`12_in z ) );

pred f is_hpartial_differentiable`21_on Z means :: PDIFF_3:def 12
( Z c= dom f & ( for z being Element of REAL 2 st z in Z holds
f | Z is_hpartial_differentiable`21_in z ) );

pred f is_hpartial_differentiable`22_on Z means :: PDIFF_3:def 13
( Z c= dom f & ( for z being Element of REAL 2 st z in Z holds
f | Z is_hpartial_differentiable`22_in z ) );

end;

:: deftheorem defines is_hpartial_differentiable`11_on PDIFF_3:def 10 :

:: deftheorem defines is_hpartial_differentiable`12_on PDIFF_3:def 11 :

:: deftheorem defines is_hpartial_differentiable`21_on PDIFF_3:def 12 :

:: deftheorem defines is_hpartial_differentiable`22_on PDIFF_3:def 13 :

definition

let f be PartFunc of (REAL 2),REAL;
let Z be set ;
assume A1: f is_hpartial_differentiable`11_on Z ;

func f `hpartial11| Z -> PartFunc of (REAL 2),REAL means :: PDIFF_3:def 14
( dom it = Z & ( for z being Element of REAL 2 st z in Z holds
it . z = hpartdiff11 (f,z) ) );

existence

proof end;

uniqueness

proof end;

end;

:: deftheorem defines `hpartial11| PDIFF_3:def 14 :

definition

let f be PartFunc of (REAL 2),REAL;
let Z be set ;
assume A1: f is_hpartial_differentiable`12_on Z ;

func f `hpartial12| Z -> PartFunc of (REAL 2),REAL means :: PDIFF_3:def 15
( dom it = Z & ( for z being Element of REAL 2 st z in Z holds
it . z = hpartdiff12 (f,z) ) );

existence

proof end;

uniqueness

proof end;

end;

:: deftheorem defines `hpartial12| PDIFF_3:def 15 :

definition

let f be PartFunc of (REAL 2),REAL;
let Z be set ;
assume A1: f is_hpartial_differentiable`21_on Z ;

func f `hpartial21| Z -> PartFunc of (REAL 2),REAL means :: PDIFF_3:def 16
( dom it = Z & ( for z being Element of REAL 2 st z in Z holds
it . z = hpartdiff21 (f,z) ) );

existence

proof end;

uniqueness

proof end;

end;

:: deftheorem defines `hpartial21| PDIFF_3:def 16 :

definition

let f be PartFunc of (REAL 2),REAL;
let Z be set ;
assume A1: f is_hpartial_differentiable`22_on Z ;

func f `hpartial22| Z -> PartFunc of (REAL 2),REAL means :: PDIFF_3:def 17
( dom it = Z & ( for z being Element of REAL 2 st z in Z holds
it . z = hpartdiff22 (f,z) ) );

existence

proof end;

uniqueness

proof end;

end;

:: deftheorem defines `hpartial22| PDIFF_3:def 17 :

theorem Th9: :: PDIFF_3:9

for z being Element of REAL 2
for f being PartFunc of (REAL 2),REAL holds
( f is_hpartial_differentiable`11_in z iff pdiff1 (f,1) is_partial_differentiable_in z,1 ) by PDIFF_2:9;

theorem Th10: :: PDIFF_3:10

for z being Element of REAL 2
for f being PartFunc of (REAL 2),REAL holds
( f is_hpartial_differentiable`12_in z iff pdiff1 (f,1) is_partial_differentiable_in z,2 ) by PDIFF_2:10;

theorem Th11: :: PDIFF_3:11

for z being Element of REAL 2
for f being PartFunc of (REAL 2),REAL holds
( f is_hpartial_differentiable`21_in z iff pdiff1 (f,2) is_partial_differentiable_in z,1 ) by PDIFF_2:9;

theorem Th12: :: PDIFF_3:12

for z being Element of REAL 2
for f being PartFunc of (REAL 2),REAL holds
( f is_hpartial_differentiable`22_in z iff pdiff1 (f,2) is_partial_differentiable_in z,2 ) by PDIFF_2:10;

theorem Th13: :: PDIFF_3:13

for z being Element of REAL 2
for f being PartFunc of (REAL 2),REAL st f is_hpartial_differentiable`11_in z holds
hpartdiff11 (f,z) = partdiff ((pdiff1 (f,1)),z,1)

proof end;

theorem Th14: :: PDIFF_3:14

for z being Element of REAL 2
for f being PartFunc of (REAL 2),REAL st f is_hpartial_differentiable`12_in z holds
hpartdiff12 (f,z) = partdiff ((pdiff1 (f,1)),z,2)

proof end;

theorem Th15: :: PDIFF_3:15

for z being Element of REAL 2
for f being PartFunc of (REAL 2),REAL st f is_hpartial_differentiable`21_in z holds
hpartdiff21 (f,z) = partdiff ((pdiff1 (f,2)),z,1)

proof end;

theorem Th16: :: PDIFF_3:16

for z being Element of REAL 2
for f being PartFunc of (REAL 2),REAL st f is_hpartial_differentiable`22_in z holds
hpartdiff22 (f,z) = partdiff ((pdiff1 (f,2)),z,2)

proof end;

theorem :: PDIFF_3:17

for f being PartFunc of (REAL 2),REAL
for z0 being Element of REAL 2
for N being Neighbourhood of (proj (1,2)) . z0 st f is_hpartial_differentiable`11_in z0 & N c= dom (SVF1 (1,(pdiff1 (f,1)),z0)) holds
for h being non-zero 0 -convergent Real_Sequence
for c being constant Real_Sequence st rng c = {((proj (1,2)) . z0)} & rng (h + c) c= N holds
( (h ") (#) (((SVF1 (1,(pdiff1 (f,1)),z0)) /* (h + c)) - ((SVF1 (1,(pdiff1 (f,1)),z0)) /* c)) is convergent & hpartdiff11 (f,z0) = lim ((h ") (#) (((SVF1 (1,(pdiff1 (f,1)),z0)) /* (h + c)) - ((SVF1 (1,(pdiff1 (f,1)),z0)) /* c))) )

proof end;

theorem :: PDIFF_3:18

for f being PartFunc of (REAL 2),REAL
for z0 being Element of REAL 2
for N being Neighbourhood of (proj (2,2)) . z0 st f is_hpartial_differentiable`12_in z0 & N c= dom (SVF1 (2,(pdiff1 (f,1)),z0)) holds
for h being non-zero 0 -convergent Real_Sequence
for c being constant Real_Sequence st rng c = {((proj (2,2)) . z0)} & rng (h + c) c= N holds
( (h ") (#) (((SVF1 (2,(pdiff1 (f,1)),z0)) /* (h + c)) - ((SVF1 (2,(pdiff1 (f,1)),z0)) /* c)) is convergent & hpartdiff12 (f,z0) = lim ((h ") (#) (((SVF1 (2,(pdiff1 (f,1)),z0)) /* (h + c)) - ((SVF1 (2,(pdiff1 (f,1)),z0)) /* c))) )

proof end;

theorem :: PDIFF_3:19

for f being PartFunc of (REAL 2),REAL
for z0 being Element of REAL 2
for N being Neighbourhood of (proj (1,2)) . z0 st f is_hpartial_differentiable`21_in z0 & N c= dom (SVF1 (1,(pdiff1 (f,2)),z0)) holds
for h being non-zero 0 -convergent Real_Sequence
for c being constant Real_Sequence st rng c = {((proj (1,2)) . z0)} & rng (h + c) c= N holds
( (h ") (#) (((SVF1 (1,(pdiff1 (f,2)),z0)) /* (h + c)) - ((SVF1 (1,(pdiff1 (f,2)),z0)) /* c)) is convergent & hpartdiff21 (f,z0) = lim ((h ") (#) (((SVF1 (1,(pdiff1 (f,2)),z0)) /* (h + c)) - ((SVF1 (1,(pdiff1 (f,2)),z0)) /* c))) )

proof end;

theorem :: PDIFF_3:20

for f being PartFunc of (REAL 2),REAL
for z0 being Element of REAL 2
for N being Neighbourhood of (proj (2,2)) . z0 st f is_hpartial_differentiable`22_in z0 & N c= dom (SVF1 (2,(pdiff1 (f,2)),z0)) holds
for h being non-zero 0 -convergent Real_Sequence
for c being constant Real_Sequence st rng c = {((proj (2,2)) . z0)} & rng (h + c) c= N holds
( (h ") (#) (((SVF1 (2,(pdiff1 (f,2)),z0)) /* (h + c)) - ((SVF1 (2,(pdiff1 (f,2)),z0)) /* c)) is convergent & hpartdiff22 (f,z0) = lim ((h ") (#) (((SVF1 (2,(pdiff1 (f,2)),z0)) /* (h + c)) - ((SVF1 (2,(pdiff1 (f,2)),z0)) /* c))) )

proof end;

theorem :: PDIFF_3:21

for z0 being Element of REAL 2
for f1, f2 being PartFunc of (REAL 2),REAL st f1 is_hpartial_differentiable`11_in z0 & f2 is_hpartial_differentiable`11_in z0 holds
( (pdiff1 (f1,1)) + (pdiff1 (f2,1)) is_partial_differentiable_in z0,1 & partdiff (((pdiff1 (f1,1)) + (pdiff1 (f2,1))),z0,1) = (hpartdiff11 (f1,z0)) + (hpartdiff11 (f2,z0)) )

proof end;

theorem :: PDIFF_3:22

for z0 being Element of REAL 2
for f1, f2 being PartFunc of (REAL 2),REAL st f1 is_hpartial_differentiable`12_in z0 & f2 is_hpartial_differentiable`12_in z0 holds
( (pdiff1 (f1,1)) + (pdiff1 (f2,1)) is_partial_differentiable_in z0,2 & partdiff (((pdiff1 (f1,1)) + (pdiff1 (f2,1))),z0,2) = (hpartdiff12 (f1,z0)) + (hpartdiff12 (f2,z0)) )

proof end;

theorem :: PDIFF_3:23

for z0 being Element of REAL 2
for f1, f2 being PartFunc of (REAL 2),REAL st f1 is_hpartial_differentiable`21_in z0 & f2 is_hpartial_differentiable`21_in z0 holds
( (pdiff1 (f1,2)) + (pdiff1 (f2,2)) is_partial_differentiable_in z0,1 & partdiff (((pdiff1 (f1,2)) + (pdiff1 (f2,2))),z0,1) = (hpartdiff21 (f1,z0)) + (hpartdiff21 (f2,z0)) )

proof end;

theorem :: PDIFF_3:24

for z0 being Element of REAL 2
for f1, f2 being PartFunc of (REAL 2),REAL st f1 is_hpartial_differentiable`22_in z0 & f2 is_hpartial_differentiable`22_in z0 holds
( (pdiff1 (f1,2)) + (pdiff1 (f2,2)) is_partial_differentiable_in z0,2 & partdiff (((pdiff1 (f1,2)) + (pdiff1 (f2,2))),z0,2) = (hpartdiff22 (f1,z0)) + (hpartdiff22 (f2,z0)) )

proof end;

theorem :: PDIFF_3:25

for z0 being Element of REAL 2
for f1, f2 being PartFunc of (REAL 2),REAL st f1 is_hpartial_differentiable`11_in z0 & f2 is_hpartial_differentiable`11_in z0 holds
( (pdiff1 (f1,1)) - (pdiff1 (f2,1)) is_partial_differentiable_in z0,1 & partdiff (((pdiff1 (f1,1)) - (pdiff1 (f2,1))),z0,1) = (hpartdiff11 (f1,z0)) - (hpartdiff11 (f2,z0)) )

proof end;

theorem :: PDIFF_3:26

for z0 being Element of REAL 2
for f1, f2 being PartFunc of (REAL 2),REAL st f1 is_hpartial_differentiable`12_in z0 & f2 is_hpartial_differentiable`12_in z0 holds
( (pdiff1 (f1,1)) - (pdiff1 (f2,1)) is_partial_differentiable_in z0,2 & partdiff (((pdiff1 (f1,1)) - (pdiff1 (f2,1))),z0,2) = (hpartdiff12 (f1,z0)) - (hpartdiff12 (f2,z0)) )

proof end;

theorem :: PDIFF_3:27

for z0 being Element of REAL 2
for f1, f2 being PartFunc of (REAL 2),REAL st f1 is_hpartial_differentiable`21_in z0 & f2 is_hpartial_differentiable`21_in z0 holds
( (pdiff1 (f1,2)) - (pdiff1 (f2,2)) is_partial_differentiable_in z0,1 & partdiff (((pdiff1 (f1,2)) - (pdiff1 (f2,2))),z0,1) = (hpartdiff21 (f1,z0)) - (hpartdiff21 (f2,z0)) )

proof end;

theorem :: PDIFF_3:28

for z0 being Element of REAL 2
for f1, f2 being PartFunc of (REAL 2),REAL st f1 is_hpartial_differentiable`22_in z0 & f2 is_hpartial_differentiable`22_in z0 holds
( (pdiff1 (f1,2)) - (pdiff1 (f2,2)) is_partial_differentiable_in z0,2 & partdiff (((pdiff1 (f1,2)) - (pdiff1 (f2,2))),z0,2) = (hpartdiff22 (f1,z0)) - (hpartdiff22 (f2,z0)) )

proof end;

theorem :: PDIFF_3:29

for r being Real
for z0 being Element of REAL 2
for f being PartFunc of (REAL 2),REAL st f is_hpartial_differentiable`11_in z0 holds
( r (#) (pdiff1 (f,1)) is_partial_differentiable_in z0,1 & partdiff ((r (#) (pdiff1 (f,1))),z0,1) = r * (hpartdiff11 (f,z0)) )

proof end;

theorem :: PDIFF_3:30

for r being Real
for z0 being Element of REAL 2
for f being PartFunc of (REAL 2),REAL st f is_hpartial_differentiable`12_in z0 holds
( r (#) (pdiff1 (f,1)) is_partial_differentiable_in z0,2 & partdiff ((r (#) (pdiff1 (f,1))),z0,2) = r * (hpartdiff12 (f,z0)) )

proof end;

theorem :: PDIFF_3:31

for r being Real
for z0 being Element of REAL 2
for f being PartFunc of (REAL 2),REAL st f is_hpartial_differentiable`21_in z0 holds
( r (#) (pdiff1 (f,2)) is_partial_differentiable_in z0,1 & partdiff ((r (#) (pdiff1 (f,2))),z0,1) = r * (hpartdiff21 (f,z0)) )

proof end;

theorem :: PDIFF_3:32

for r being Real
for z0 being Element of REAL 2
for f being PartFunc of (REAL 2),REAL st f is_hpartial_differentiable`22_in z0 holds
( r (#) (pdiff1 (f,2)) is_partial_differentiable_in z0,2 & partdiff ((r (#) (pdiff1 (f,2))),z0,2) = r * (hpartdiff22 (f,z0)) )

proof end;

theorem :: PDIFF_3:33

for z0 being Element of REAL 2
for f1, f2 being PartFunc of (REAL 2),REAL st f1 is_hpartial_differentiable`11_in z0 & f2 is_hpartial_differentiable`11_in z0 holds
(pdiff1 (f1,1)) (#) (pdiff1 (f2,1)) is_partial_differentiable_in z0,1

proof end;

theorem :: PDIFF_3:34

for z0 being Element of REAL 2
for f1, f2 being PartFunc of (REAL 2),REAL st f1 is_hpartial_differentiable`12_in z0 & f2 is_hpartial_differentiable`12_in z0 holds
(pdiff1 (f1,1)) (#) (pdiff1 (f2,1)) is_partial_differentiable_in z0,2

proof end;

theorem :: PDIFF_3:35

for z0 being Element of REAL 2
for f1, f2 being PartFunc of (REAL 2),REAL st f1 is_hpartial_differentiable`21_in z0 & f2 is_hpartial_differentiable`21_in z0 holds
(pdiff1 (f1,2)) (#) (pdiff1 (f2,2)) is_partial_differentiable_in z0,1

proof end;

theorem :: PDIFF_3:36

for z0 being Element of REAL 2
for f1, f2 being PartFunc of (REAL 2),REAL st f1 is_hpartial_differentiable`22_in z0 & f2 is_hpartial_differentiable`22_in z0 holds
(pdiff1 (f1,2)) (#) (pdiff1 (f2,2)) is_partial_differentiable_in z0,2

proof end;

theorem :: PDIFF_3:37

for f being PartFunc of (REAL 2),REAL
for z0 being Element of REAL 2 st f is_hpartial_differentiable`11_in z0 holds
SVF1 (1,(pdiff1 (f,1)),z0) is_continuous_in (proj (1,2)) . z0 by Th9, PDIFF_2:21;

theorem :: PDIFF_3:38

for f being PartFunc of (REAL 2),REAL
for z0 being Element of REAL 2 st f is_hpartial_differentiable`12_in z0 holds
SVF1 (2,(pdiff1 (f,1)),z0) is_continuous_in (proj (2,2)) . z0 by Th10, PDIFF_2:22;

theorem :: PDIFF_3:39

for f being PartFunc of (REAL 2),REAL
for z0 being Element of REAL 2 st f is_hpartial_differentiable`21_in z0 holds
SVF1 (1,(pdiff1 (f,2)),z0) is_continuous_in (proj (1,2)) . z0 by Th11, PDIFF_2:21;

theorem :: PDIFF_3:40

for f being PartFunc of (REAL 2),REAL
for z0 being Element of REAL 2 st f is_hpartial_differentiable`22_in z0 holds
SVF1 (2,(pdiff1 (f,2)),z0) is_continuous_in (proj (2,2)) . z0 by Th12, PDIFF_2:22;