PDIFF_5: Second-order Partial Differentiation of Real Ternary Functions

:: Second-order Partial Differentiation of Real Ternary Functions
:: by Takao Inou\'e
::
:: Received January 26, 2010
:: Copyright (c) 2010-2021 Association of Mizar Users

definition

let f be PartFunc of (REAL 3),REAL;
let u be Element of REAL 3;

pred f is_hpartial_differentiable`11_in u means :: PDIFF_5:def 1
ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of x0 st
( N c= dom (SVF1 (1,(pdiff1 (f,1)),u)) & ex L being LinearFunc ex R being RestFunc st
for x being Real st x in N holds
((SVF1 (1,(pdiff1 (f,1)),u)) . x) - ((SVF1 (1,(pdiff1 (f,1)),u)) . x0) = (L . (x - x0)) + (R . (x - x0)) ) );

pred f is_hpartial_differentiable`12_in u means :: PDIFF_5:def 2
ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of y0 st
( N c= dom (SVF1 (2,(pdiff1 (f,1)),u)) & ex L being LinearFunc ex R being RestFunc st
for y being Real st y in N holds
((SVF1 (2,(pdiff1 (f,1)),u)) . y) - ((SVF1 (2,(pdiff1 (f,1)),u)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) );

pred f is_hpartial_differentiable`13_in u means :: PDIFF_5:def 3
ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of z0 st
( N c= dom (SVF1 (3,(pdiff1 (f,1)),u)) & ex L being LinearFunc ex R being RestFunc st
for z being Real st z in N holds
((SVF1 (3,(pdiff1 (f,1)),u)) . z) - ((SVF1 (3,(pdiff1 (f,1)),u)) . z0) = (L . (z - z0)) + (R . (z - z0)) ) );

pred f is_hpartial_differentiable`21_in u means :: PDIFF_5:def 4
ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of x0 st
( N c= dom (SVF1 (1,(pdiff1 (f,2)),u)) & ex L being LinearFunc ex R being RestFunc st
for x being Real st x in N holds
((SVF1 (1,(pdiff1 (f,2)),u)) . x) - ((SVF1 (1,(pdiff1 (f,2)),u)) . x0) = (L . (x - x0)) + (R . (x - x0)) ) );

pred f is_hpartial_differentiable`22_in u means :: PDIFF_5:def 5
ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of y0 st
( N c= dom (SVF1 (2,(pdiff1 (f,2)),u)) & ex L being LinearFunc ex R being RestFunc st
for y being Real st y in N holds
((SVF1 (2,(pdiff1 (f,2)),u)) . y) - ((SVF1 (2,(pdiff1 (f,2)),u)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) );

pred f is_hpartial_differentiable`23_in u means :: PDIFF_5:def 6
ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of z0 st
( N c= dom (SVF1 (3,(pdiff1 (f,2)),u)) & ex L being LinearFunc ex R being RestFunc st
for z being Real st z in N holds
((SVF1 (3,(pdiff1 (f,2)),u)) . z) - ((SVF1 (3,(pdiff1 (f,2)),u)) . z0) = (L . (z - z0)) + (R . (z - z0)) ) );

pred f is_hpartial_differentiable`31_in u means :: PDIFF_5:def 7
ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of x0 st
( N c= dom (SVF1 (1,(pdiff1 (f,3)),u)) & ex L being LinearFunc ex R being RestFunc st
for x being Real st x in N holds
((SVF1 (1,(pdiff1 (f,3)),u)) . x) - ((SVF1 (1,(pdiff1 (f,3)),u)) . x0) = (L . (x - x0)) + (R . (x - x0)) ) );

pred f is_hpartial_differentiable`32_in u means :: PDIFF_5:def 8
ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of y0 st
( N c= dom (SVF1 (2,(pdiff1 (f,3)),u)) & ex L being LinearFunc ex R being RestFunc st
for y being Real st y in N holds
((SVF1 (2,(pdiff1 (f,3)),u)) . y) - ((SVF1 (2,(pdiff1 (f,3)),u)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) );

pred f is_hpartial_differentiable`33_in u means :: PDIFF_5:def 9
ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of z0 st
( N c= dom (SVF1 (3,(pdiff1 (f,3)),u)) & ex L being LinearFunc ex R being RestFunc st
for z being Real st z in N holds
((SVF1 (3,(pdiff1 (f,3)),u)) . z) - ((SVF1 (3,(pdiff1 (f,3)),u)) . z0) = (L . (z - z0)) + (R . (z - z0)) ) );

end;

:: deftheorem defines is_hpartial_differentiable`11_in PDIFF_5:def 1 :

:: deftheorem defines is_hpartial_differentiable`12_in PDIFF_5:def 2 :

:: deftheorem defines is_hpartial_differentiable`13_in PDIFF_5:def 3 :

:: deftheorem defines is_hpartial_differentiable`21_in PDIFF_5:def 4 :

:: deftheorem defines is_hpartial_differentiable`22_in PDIFF_5:def 5 :

:: deftheorem defines is_hpartial_differentiable`23_in PDIFF_5:def 6 :

:: deftheorem defines is_hpartial_differentiable`31_in PDIFF_5:def 7 :

:: deftheorem defines is_hpartial_differentiable`32_in PDIFF_5:def 8 :

:: deftheorem defines is_hpartial_differentiable`33_in PDIFF_5:def 9 :

definition

let f be PartFunc of (REAL 3),REAL;
let u be Element of REAL 3;
assume A1: f is_hpartial_differentiable`11_in u ;

func hpartdiff11 (f,u) -> Real means :Def10: :: PDIFF_5:def 10
ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of x0 st
( N c= dom (SVF1 (1,(pdiff1 (f,1)),u)) & 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)),u)) . x) - ((SVF1 (1,(pdiff1 (f,1)),u)) . x0) = (L . (x - x0)) + (R . (x - x0)) ) ) ) );

proof end;

proof end;

end;

:: deftheorem Def10 defines hpartdiff11 PDIFF_5:def 10 :

definition

let f be PartFunc of (REAL 3),REAL;
let u be Element of REAL 3;
assume A1: f is_hpartial_differentiable`12_in u ;

func hpartdiff12 (f,u) -> Real means :Def11: :: PDIFF_5:def 11
ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of y0 st
( N c= dom (SVF1 (2,(pdiff1 (f,1)),u)) & 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)),u)) . y) - ((SVF1 (2,(pdiff1 (f,1)),u)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) ) );

proof end;

proof end;

end;

:: deftheorem Def11 defines hpartdiff12 PDIFF_5:def 11 :

definition

let f be PartFunc of (REAL 3),REAL;
let u be Element of REAL 3;
assume A1: f is_hpartial_differentiable`13_in u ;

func hpartdiff13 (f,u) -> Real means :Def12: :: PDIFF_5:def 12
ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of z0 st
( N c= dom (SVF1 (3,(pdiff1 (f,1)),u)) & ex L being LinearFunc ex R being RestFunc st
( it = L . 1 & ( for z being Real st z in N holds
((SVF1 (3,(pdiff1 (f,1)),u)) . z) - ((SVF1 (3,(pdiff1 (f,1)),u)) . z0) = (L . (z - z0)) + (R . (z - z0)) ) ) ) );

proof end;

proof end;

end;

:: deftheorem Def12 defines hpartdiff13 PDIFF_5:def 12 :

definition

let f be PartFunc of (REAL 3),REAL;
let u be Element of REAL 3;
assume A1: f is_hpartial_differentiable`21_in u ;

func hpartdiff21 (f,u) -> Real means :Def13: :: PDIFF_5:def 13
ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of x0 st
( N c= dom (SVF1 (1,(pdiff1 (f,2)),u)) & 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)),u)) . x) - ((SVF1 (1,(pdiff1 (f,2)),u)) . x0) = (L . (x - x0)) + (R . (x - x0)) ) ) ) );

proof end;

proof end;

end;

:: deftheorem Def13 defines hpartdiff21 PDIFF_5:def 13 :

definition

let f be PartFunc of (REAL 3),REAL;
let u be Element of REAL 3;
assume A1: f is_hpartial_differentiable`22_in u ;

func hpartdiff22 (f,u) -> Real means :Def14: :: PDIFF_5:def 14
ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of y0 st
( N c= dom (SVF1 (2,(pdiff1 (f,2)),u)) & 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)),u)) . y) - ((SVF1 (2,(pdiff1 (f,2)),u)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) ) );

proof end;

proof end;

end;

:: deftheorem Def14 defines hpartdiff22 PDIFF_5:def 14 :

definition

let f be PartFunc of (REAL 3),REAL;
let u be Element of REAL 3;
assume A1: f is_hpartial_differentiable`23_in u ;

func hpartdiff23 (f,u) -> Real means :Def15: :: PDIFF_5:def 15
ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of z0 st
( N c= dom (SVF1 (3,(pdiff1 (f,2)),u)) & ex L being LinearFunc ex R being RestFunc st
( it = L . 1 & ( for z being Real st z in N holds
((SVF1 (3,(pdiff1 (f,2)),u)) . z) - ((SVF1 (3,(pdiff1 (f,2)),u)) . z0) = (L . (z - z0)) + (R . (z - z0)) ) ) ) );

proof end;

proof end;

end;

:: deftheorem Def15 defines hpartdiff23 PDIFF_5:def 15 :

definition

let f be PartFunc of (REAL 3),REAL;
let u be Element of REAL 3;
assume A1: f is_hpartial_differentiable`31_in u ;

func hpartdiff31 (f,u) -> Real means :Def16: :: PDIFF_5:def 16
ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of x0 st
( N c= dom (SVF1 (1,(pdiff1 (f,3)),u)) & 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,3)),u)) . x) - ((SVF1 (1,(pdiff1 (f,3)),u)) . x0) = (L . (x - x0)) + (R . (x - x0)) ) ) ) );

proof end;

proof end;

end;

:: deftheorem Def16 defines hpartdiff31 PDIFF_5:def 16 :

definition

let f be PartFunc of (REAL 3),REAL;
let u be Element of REAL 3;
assume A1: f is_hpartial_differentiable`32_in u ;

func hpartdiff32 (f,u) -> Real means :Def17: :: PDIFF_5:def 17
ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of y0 st
( N c= dom (SVF1 (2,(pdiff1 (f,3)),u)) & 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,3)),u)) . y) - ((SVF1 (2,(pdiff1 (f,3)),u)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) ) );

proof end;

proof end;

end;

:: deftheorem Def17 defines hpartdiff32 PDIFF_5:def 17 :

definition

let f be PartFunc of (REAL 3),REAL;
let u be Element of REAL 3;
assume A1: f is_hpartial_differentiable`33_in u ;

func hpartdiff33 (f,u) -> Real means :Def18: :: PDIFF_5:def 18
ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of z0 st
( N c= dom (SVF1 (3,(pdiff1 (f,3)),u)) & ex L being LinearFunc ex R being RestFunc st
( it = L . 1 & ( for z being Real st z in N holds
((SVF1 (3,(pdiff1 (f,3)),u)) . z) - ((SVF1 (3,(pdiff1 (f,3)),u)) . z0) = (L . (z - z0)) + (R . (z - z0)) ) ) ) );

proof end;

proof end;

end;

:: deftheorem Def18 defines hpartdiff33 PDIFF_5:def 18 :

theorem :: PDIFF_5:1

for x0, y0, z0 being Real
for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st u = <*x0,y0,z0*> & f is_hpartial_differentiable`11_in u holds
SVF1 (1,(pdiff1 (f,1)),u) is_differentiable_in x0

proof end;

theorem :: PDIFF_5:2

for x0, y0, z0 being Real
for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st u = <*x0,y0,z0*> & f is_hpartial_differentiable`12_in u holds
SVF1 (2,(pdiff1 (f,1)),u) is_differentiable_in y0

proof end;

theorem :: PDIFF_5:3

for x0, y0, z0 being Real
for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st u = <*x0,y0,z0*> & f is_hpartial_differentiable`13_in u holds
SVF1 (3,(pdiff1 (f,1)),u) is_differentiable_in z0

proof end;

theorem :: PDIFF_5:4

for x0, y0, z0 being Real
for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st u = <*x0,y0,z0*> & f is_hpartial_differentiable`21_in u holds
SVF1 (1,(pdiff1 (f,2)),u) is_differentiable_in x0

proof end;

theorem :: PDIFF_5:5

for x0, y0, z0 being Real
for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st u = <*x0,y0,z0*> & f is_hpartial_differentiable`22_in u holds
SVF1 (2,(pdiff1 (f,2)),u) is_differentiable_in y0

proof end;

theorem :: PDIFF_5:6

for x0, y0, z0 being Real
for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st u = <*x0,y0,z0*> & f is_hpartial_differentiable`23_in u holds
SVF1 (3,(pdiff1 (f,2)),u) is_differentiable_in z0

proof end;

theorem :: PDIFF_5:7

for x0, y0, z0 being Real
for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st u = <*x0,y0,z0*> & f is_hpartial_differentiable`31_in u holds
SVF1 (1,(pdiff1 (f,3)),u) is_differentiable_in x0

proof end;

theorem :: PDIFF_5:8

for x0, y0, z0 being Real
for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st u = <*x0,y0,z0*> & f is_hpartial_differentiable`32_in u holds
SVF1 (2,(pdiff1 (f,3)),u) is_differentiable_in y0

proof end;

theorem :: PDIFF_5:9

for x0, y0, z0 being Real
for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st u = <*x0,y0,z0*> & f is_hpartial_differentiable`33_in u holds
SVF1 (3,(pdiff1 (f,3)),u) is_differentiable_in z0

proof end;

theorem Th10: :: PDIFF_5:10

for x0, y0, z0 being Real
for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st u = <*x0,y0,z0*> & f is_hpartial_differentiable`11_in u holds
hpartdiff11 (f,u) = diff ((SVF1 (1,(pdiff1 (f,1)),u)),x0)

proof end;

theorem Th11: :: PDIFF_5:11

for x0, y0, z0 being Real
for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st u = <*x0,y0,z0*> & f is_hpartial_differentiable`12_in u holds
hpartdiff12 (f,u) = diff ((SVF1 (2,(pdiff1 (f,1)),u)),y0)

proof end;

theorem Th12: :: PDIFF_5:12

for x0, y0, z0 being Real
for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st u = <*x0,y0,z0*> & f is_hpartial_differentiable`13_in u holds
hpartdiff13 (f,u) = diff ((SVF1 (3,(pdiff1 (f,1)),u)),z0)

proof end;

theorem Th13: :: PDIFF_5:13

for x0, y0, z0 being Real
for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st u = <*x0,y0,z0*> & f is_hpartial_differentiable`21_in u holds
hpartdiff21 (f,u) = diff ((SVF1 (1,(pdiff1 (f,2)),u)),x0)

proof end;

theorem Th14: :: PDIFF_5:14

for x0, y0, z0 being Real
for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st u = <*x0,y0,z0*> & f is_hpartial_differentiable`22_in u holds
hpartdiff22 (f,u) = diff ((SVF1 (2,(pdiff1 (f,2)),u)),y0)

proof end;

theorem Th15: :: PDIFF_5:15

for x0, y0, z0 being Real
for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st u = <*x0,y0,z0*> & f is_hpartial_differentiable`23_in u holds
hpartdiff23 (f,u) = diff ((SVF1 (3,(pdiff1 (f,2)),u)),z0)

proof end;

theorem Th16: :: PDIFF_5:16

for x0, y0, z0 being Real
for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st u = <*x0,y0,z0*> & f is_hpartial_differentiable`31_in u holds
hpartdiff31 (f,u) = diff ((SVF1 (1,(pdiff1 (f,3)),u)),x0)

proof end;

theorem Th17: :: PDIFF_5:17

for x0, y0, z0 being Real
for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st u = <*x0,y0,z0*> & f is_hpartial_differentiable`32_in u holds
hpartdiff32 (f,u) = diff ((SVF1 (2,(pdiff1 (f,3)),u)),y0)

proof end;

theorem Th18: :: PDIFF_5:18

for x0, y0, z0 being Real
for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st u = <*x0,y0,z0*> & f is_hpartial_differentiable`33_in u holds
hpartdiff33 (f,u) = diff ((SVF1 (3,(pdiff1 (f,3)),u)),z0)

proof end;

definition

let f be PartFunc of (REAL 3),REAL;
let D be set ;

pred f is_hpartial_differentiable`11_on D means :: PDIFF_5:def 19
( D c= dom f & ( for u being Element of REAL 3 st u in D holds
f | D is_hpartial_differentiable`11_in u ) );

pred f is_hpartial_differentiable`12_on D means :: PDIFF_5:def 20
( D c= dom f & ( for u being Element of REAL 3 st u in D holds
f | D is_hpartial_differentiable`12_in u ) );

pred f is_hpartial_differentiable`13_on D means :: PDIFF_5:def 21
( D c= dom f & ( for u being Element of REAL 3 st u in D holds
f | D is_hpartial_differentiable`13_in u ) );

pred f is_hpartial_differentiable`21_on D means :: PDIFF_5:def 22
( D c= dom f & ( for u being Element of REAL 3 st u in D holds
f | D is_hpartial_differentiable`21_in u ) );

pred f is_hpartial_differentiable`22_on D means :: PDIFF_5:def 23
( D c= dom f & ( for u being Element of REAL 3 st u in D holds
f | D is_hpartial_differentiable`22_in u ) );

pred f is_hpartial_differentiable`23_on D means :: PDIFF_5:def 24
( D c= dom f & ( for u being Element of REAL 3 st u in D holds
f | D is_hpartial_differentiable`23_in u ) );

pred f is_hpartial_differentiable`31_on D means :: PDIFF_5:def 25
( D c= dom f & ( for u being Element of REAL 3 st u in D holds
f | D is_hpartial_differentiable`31_in u ) );

pred f is_hpartial_differentiable`32_on D means :: PDIFF_5:def 26
( D c= dom f & ( for u being Element of REAL 3 st u in D holds
f | D is_hpartial_differentiable`32_in u ) );

pred f is_hpartial_differentiable`33_on D means :: PDIFF_5:def 27
( D c= dom f & ( for u being Element of REAL 3 st u in D holds
f | D is_hpartial_differentiable`33_in u ) );

end;

:: deftheorem defines is_hpartial_differentiable`11_on PDIFF_5:def 19 :

:: deftheorem defines is_hpartial_differentiable`12_on PDIFF_5:def 20 :

:: deftheorem defines is_hpartial_differentiable`13_on PDIFF_5:def 21 :

:: deftheorem defines is_hpartial_differentiable`21_on PDIFF_5:def 22 :

:: deftheorem defines is_hpartial_differentiable`22_on PDIFF_5:def 23 :

:: deftheorem defines is_hpartial_differentiable`23_on PDIFF_5:def 24 :

:: deftheorem defines is_hpartial_differentiable`31_on PDIFF_5:def 25 :

:: deftheorem defines is_hpartial_differentiable`32_on PDIFF_5:def 26 :

:: deftheorem defines is_hpartial_differentiable`33_on PDIFF_5:def 27 :

definition

let f be PartFunc of (REAL 3),REAL;
let D be set ;
assume A1: f is_hpartial_differentiable`11_on D ;

func f `hpartial11| D -> PartFunc of (REAL 3),REAL means :: PDIFF_5:def 28
( dom it = D & ( for u being Element of REAL 3 st u in D holds
it . u = hpartdiff11 (f,u) ) );

proof end;

proof end;

end;

:: deftheorem defines `hpartial11| PDIFF_5:def 28 :

definition

let f be PartFunc of (REAL 3),REAL;
let D be set ;
assume A1: f is_hpartial_differentiable`12_on D ;

func f `hpartial12| D -> PartFunc of (REAL 3),REAL means :: PDIFF_5:def 29
( dom it = D & ( for u being Element of REAL 3 st u in D holds
it . u = hpartdiff12 (f,u) ) );

proof end;

proof end;

end;

:: deftheorem defines `hpartial12| PDIFF_5:def 29 :

definition

let f be PartFunc of (REAL 3),REAL;
let D be set ;
assume A1: f is_hpartial_differentiable`13_on D ;

func f `hpartial13| D -> PartFunc of (REAL 3),REAL means :: PDIFF_5:def 30
( dom it = D & ( for u being Element of REAL 3 st u in D holds
it . u = hpartdiff13 (f,u) ) );

proof end;

proof end;

end;

:: deftheorem defines `hpartial13| PDIFF_5:def 30 :

definition

let f be PartFunc of (REAL 3),REAL;
let D be set ;
assume A1: f is_hpartial_differentiable`21_on D ;

func f `hpartial21| D -> PartFunc of (REAL 3),REAL means :: PDIFF_5:def 31
( dom it = D & ( for u being Element of REAL 3 st u in D holds
it . u = hpartdiff21 (f,u) ) );

proof end;

proof end;

end;

:: deftheorem defines `hpartial21| PDIFF_5:def 31 :

definition

let f be PartFunc of (REAL 3),REAL;
let D be set ;
assume A1: f is_hpartial_differentiable`22_on D ;

func f `hpartial22| D -> PartFunc of (REAL 3),REAL means :: PDIFF_5:def 32
( dom it = D & ( for u being Element of REAL 3 st u in D holds
it . u = hpartdiff22 (f,u) ) );

proof end;

proof end;

end;

:: deftheorem defines `hpartial22| PDIFF_5:def 32 :

definition

let f be PartFunc of (REAL 3),REAL;
let D be set ;
assume A1: f is_hpartial_differentiable`23_on D ;

func f `hpartial23| D -> PartFunc of (REAL 3),REAL means :: PDIFF_5:def 33
( dom it = D & ( for u being Element of REAL 3 st u in D holds
it . u = hpartdiff23 (f,u) ) );

proof end;

proof end;

end;

:: deftheorem defines `hpartial23| PDIFF_5:def 33 :

definition

let f be PartFunc of (REAL 3),REAL;
let D be set ;
assume A1: f is_hpartial_differentiable`31_on D ;

func f `hpartial31| D -> PartFunc of (REAL 3),REAL means :: PDIFF_5:def 34
( dom it = D & ( for u being Element of REAL 3 st u in D holds
it . u = hpartdiff31 (f,u) ) );

proof end;

proof end;

end;

:: deftheorem defines `hpartial31| PDIFF_5:def 34 :

definition

let f be PartFunc of (REAL 3),REAL;
let D be set ;
assume A1: f is_hpartial_differentiable`32_on D ;

func f `hpartial32| D -> PartFunc of (REAL 3),REAL means :: PDIFF_5:def 35
( dom it = D & ( for u being Element of REAL 3 st u in D holds
it . u = hpartdiff32 (f,u) ) );

proof end;

proof end;

end;

:: deftheorem defines `hpartial32| PDIFF_5:def 35 :

definition

let f be PartFunc of (REAL 3),REAL;
let D be set ;
assume A1: f is_hpartial_differentiable`33_on D ;

func f `hpartial33| D -> PartFunc of (REAL 3),REAL means :: PDIFF_5:def 36
( dom it = D & ( for u being Element of REAL 3 st u in D holds
it . u = hpartdiff33 (f,u) ) );

proof end;

proof end;

end;

:: deftheorem defines `hpartial33| PDIFF_5:def 36 :

theorem Th19: :: PDIFF_5:19

for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL holds
( f is_hpartial_differentiable`11_in u iff pdiff1 (f,1) is_partial_differentiable_in u,1 ) by PDIFF_4:13;

theorem Th20: :: PDIFF_5:20

for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL holds
( f is_hpartial_differentiable`12_in u iff pdiff1 (f,1) is_partial_differentiable_in u,2 ) by PDIFF_4:14;

theorem Th21: :: PDIFF_5:21

for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL holds
( f is_hpartial_differentiable`13_in u iff pdiff1 (f,1) is_partial_differentiable_in u,3 ) by PDIFF_4:15;

theorem Th22: :: PDIFF_5:22

for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL holds
( f is_hpartial_differentiable`21_in u iff pdiff1 (f,2) is_partial_differentiable_in u,1 ) by PDIFF_4:13;

theorem Th23: :: PDIFF_5:23

for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL holds
( f is_hpartial_differentiable`22_in u iff pdiff1 (f,2) is_partial_differentiable_in u,2 ) by PDIFF_4:14;

theorem Th24: :: PDIFF_5:24

for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL holds
( f is_hpartial_differentiable`23_in u iff pdiff1 (f,2) is_partial_differentiable_in u,3 ) by PDIFF_4:15;

theorem Th25: :: PDIFF_5:25

for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL holds
( f is_hpartial_differentiable`31_in u iff pdiff1 (f,3) is_partial_differentiable_in u,1 ) by PDIFF_4:13;

theorem Th26: :: PDIFF_5:26

for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL holds
( f is_hpartial_differentiable`32_in u iff pdiff1 (f,3) is_partial_differentiable_in u,2 ) by PDIFF_4:14;

theorem Th27: :: PDIFF_5:27

for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL holds
( f is_hpartial_differentiable`33_in u iff pdiff1 (f,3) is_partial_differentiable_in u,3 ) by PDIFF_4:15;

theorem Th28: :: PDIFF_5:28

for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st f is_hpartial_differentiable`11_in u holds
hpartdiff11 (f,u) = partdiff ((pdiff1 (f,1)),u,1)

proof end;

theorem Th29: :: PDIFF_5:29

for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st f is_hpartial_differentiable`12_in u holds
hpartdiff12 (f,u) = partdiff ((pdiff1 (f,1)),u,2)

proof end;

theorem Th30: :: PDIFF_5:30

for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st f is_hpartial_differentiable`13_in u holds
hpartdiff13 (f,u) = partdiff ((pdiff1 (f,1)),u,3)

proof end;

theorem Th31: :: PDIFF_5:31

for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st f is_hpartial_differentiable`21_in u holds
hpartdiff21 (f,u) = partdiff ((pdiff1 (f,2)),u,1)

proof end;

theorem Th32: :: PDIFF_5:32

for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st f is_hpartial_differentiable`22_in u holds
hpartdiff22 (f,u) = partdiff ((pdiff1 (f,2)),u,2)

proof end;

theorem Th33: :: PDIFF_5:33

for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st f is_hpartial_differentiable`23_in u holds
hpartdiff23 (f,u) = partdiff ((pdiff1 (f,2)),u,3)

proof end;

theorem Th34: :: PDIFF_5:34

for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st f is_hpartial_differentiable`31_in u holds
hpartdiff31 (f,u) = partdiff ((pdiff1 (f,3)),u,1)

proof end;

theorem Th35: :: PDIFF_5:35

for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st f is_hpartial_differentiable`32_in u holds
hpartdiff32 (f,u) = partdiff ((pdiff1 (f,3)),u,2)

proof end;

theorem Th36: :: PDIFF_5:36

for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st f is_hpartial_differentiable`33_in u holds
hpartdiff33 (f,u) = partdiff ((pdiff1 (f,3)),u,3)

proof end;

theorem :: PDIFF_5:37

for f being PartFunc of (REAL 3),REAL
for u0 being Element of REAL 3
for N being Neighbourhood of (proj (1,3)) . u0 st f is_hpartial_differentiable`11_in u0 & N c= dom (SVF1 (1,(pdiff1 (f,1)),u0)) holds
for h being non-zero 0 -convergent Real_Sequence
for c being constant Real_Sequence st rng c = {((proj (1,3)) . u0)} & rng (h + c) c= N holds
( (h ") (#) (((SVF1 (1,(pdiff1 (f,1)),u0)) /* (h + c)) - ((SVF1 (1,(pdiff1 (f,1)),u0)) /* c)) is convergent & hpartdiff11 (f,u0) = lim ((h ") (#) (((SVF1 (1,(pdiff1 (f,1)),u0)) /* (h + c)) - ((SVF1 (1,(pdiff1 (f,1)),u0)) /* c))) )

proof end;

theorem :: PDIFF_5:38

for f being PartFunc of (REAL 3),REAL
for u0 being Element of REAL 3
for N being Neighbourhood of (proj (2,3)) . u0 st f is_hpartial_differentiable`12_in u0 & N c= dom (SVF1 (2,(pdiff1 (f,1)),u0)) holds
for h being non-zero 0 -convergent Real_Sequence
for c being constant Real_Sequence st rng c = {((proj (2,3)) . u0)} & rng (h + c) c= N holds
( (h ") (#) (((SVF1 (2,(pdiff1 (f,1)),u0)) /* (h + c)) - ((SVF1 (2,(pdiff1 (f,1)),u0)) /* c)) is convergent & hpartdiff12 (f,u0) = lim ((h ") (#) (((SVF1 (2,(pdiff1 (f,1)),u0)) /* (h + c)) - ((SVF1 (2,(pdiff1 (f,1)),u0)) /* c))) )

proof end;

theorem :: PDIFF_5:39

for f being PartFunc of (REAL 3),REAL
for u0 being Element of REAL 3
for N being Neighbourhood of (proj (3,3)) . u0 st f is_hpartial_differentiable`13_in u0 & N c= dom (SVF1 (3,(pdiff1 (f,1)),u0)) holds
for h being non-zero 0 -convergent Real_Sequence
for c being constant Real_Sequence st rng c = {((proj (3,3)) . u0)} & rng (h + c) c= N holds
( (h ") (#) (((SVF1 (3,(pdiff1 (f,1)),u0)) /* (h + c)) - ((SVF1 (3,(pdiff1 (f,1)),u0)) /* c)) is convergent & hpartdiff13 (f,u0) = lim ((h ") (#) (((SVF1 (3,(pdiff1 (f,1)),u0)) /* (h + c)) - ((SVF1 (3,(pdiff1 (f,1)),u0)) /* c))) )

proof end;

theorem :: PDIFF_5:40

for f being PartFunc of (REAL 3),REAL
for u0 being Element of REAL 3
for N being Neighbourhood of (proj (1,3)) . u0 st f is_hpartial_differentiable`21_in u0 & N c= dom (SVF1 (1,(pdiff1 (f,2)),u0)) holds
for h being non-zero 0 -convergent Real_Sequence
for c being constant Real_Sequence st rng c = {((proj (1,3)) . u0)} & rng (h + c) c= N holds
( (h ") (#) (((SVF1 (1,(pdiff1 (f,2)),u0)) /* (h + c)) - ((SVF1 (1,(pdiff1 (f,2)),u0)) /* c)) is convergent & hpartdiff21 (f,u0) = lim ((h ") (#) (((SVF1 (1,(pdiff1 (f,2)),u0)) /* (h + c)) - ((SVF1 (1,(pdiff1 (f,2)),u0)) /* c))) )

proof end;

theorem :: PDIFF_5:41

for f being PartFunc of (REAL 3),REAL
for u0 being Element of REAL 3
for N being Neighbourhood of (proj (2,3)) . u0 st f is_hpartial_differentiable`22_in u0 & N c= dom (SVF1 (2,(pdiff1 (f,2)),u0)) holds
for h being non-zero 0 -convergent Real_Sequence
for c being constant Real_Sequence st rng c = {((proj (2,3)) . u0)} & rng (h + c) c= N holds
( (h ") (#) (((SVF1 (2,(pdiff1 (f,2)),u0)) /* (h + c)) - ((SVF1 (2,(pdiff1 (f,2)),u0)) /* c)) is convergent & hpartdiff22 (f,u0) = lim ((h ") (#) (((SVF1 (2,(pdiff1 (f,2)),u0)) /* (h + c)) - ((SVF1 (2,(pdiff1 (f,2)),u0)) /* c))) )

proof end;

theorem :: PDIFF_5:42

for f being PartFunc of (REAL 3),REAL
for u0 being Element of REAL 3
for N being Neighbourhood of (proj (3,3)) . u0 st f is_hpartial_differentiable`23_in u0 & N c= dom (SVF1 (3,(pdiff1 (f,2)),u0)) holds
for h being non-zero 0 -convergent Real_Sequence
for c being constant Real_Sequence st rng c = {((proj (3,3)) . u0)} & rng (h + c) c= N holds
( (h ") (#) (((SVF1 (3,(pdiff1 (f,2)),u0)) /* (h + c)) - ((SVF1 (3,(pdiff1 (f,2)),u0)) /* c)) is convergent & hpartdiff23 (f,u0) = lim ((h ") (#) (((SVF1 (3,(pdiff1 (f,2)),u0)) /* (h + c)) - ((SVF1 (3,(pdiff1 (f,2)),u0)) /* c))) )

proof end;

theorem :: PDIFF_5:43

for f being PartFunc of (REAL 3),REAL
for u0 being Element of REAL 3
for N being Neighbourhood of (proj (1,3)) . u0 st f is_hpartial_differentiable`31_in u0 & N c= dom (SVF1 (1,(pdiff1 (f,3)),u0)) holds
for h being non-zero 0 -convergent Real_Sequence
for c being constant Real_Sequence st rng c = {((proj (1,3)) . u0)} & rng (h + c) c= N holds
( (h ") (#) (((SVF1 (1,(pdiff1 (f,3)),u0)) /* (h + c)) - ((SVF1 (1,(pdiff1 (f,3)),u0)) /* c)) is convergent & hpartdiff31 (f,u0) = lim ((h ") (#) (((SVF1 (1,(pdiff1 (f,3)),u0)) /* (h + c)) - ((SVF1 (1,(pdiff1 (f,3)),u0)) /* c))) )

proof end;

theorem :: PDIFF_5:44

for f being PartFunc of (REAL 3),REAL
for u0 being Element of REAL 3
for N being Neighbourhood of (proj (2,3)) . u0 st f is_hpartial_differentiable`32_in u0 & N c= dom (SVF1 (2,(pdiff1 (f,3)),u0)) holds
for h being non-zero 0 -convergent Real_Sequence
for c being constant Real_Sequence st rng c = {((proj (2,3)) . u0)} & rng (h + c) c= N holds
( (h ") (#) (((SVF1 (2,(pdiff1 (f,3)),u0)) /* (h + c)) - ((SVF1 (2,(pdiff1 (f,3)),u0)) /* c)) is convergent & hpartdiff32 (f,u0) = lim ((h ") (#) (((SVF1 (2,(pdiff1 (f,3)),u0)) /* (h + c)) - ((SVF1 (2,(pdiff1 (f,3)),u0)) /* c))) )

proof end;

theorem :: PDIFF_5:45

for f being PartFunc of (REAL 3),REAL
for u0 being Element of REAL 3
for N being Neighbourhood of (proj (3,3)) . u0 st f is_hpartial_differentiable`33_in u0 & N c= dom (SVF1 (3,(pdiff1 (f,3)),u0)) holds
for h being non-zero 0 -convergent Real_Sequence
for c being constant Real_Sequence st rng c = {((proj (3,3)) . u0)} & rng (h + c) c= N holds
( (h ") (#) (((SVF1 (3,(pdiff1 (f,3)),u0)) /* (h + c)) - ((SVF1 (3,(pdiff1 (f,3)),u0)) /* c)) is convergent & hpartdiff33 (f,u0) = lim ((h ") (#) (((SVF1 (3,(pdiff1 (f,3)),u0)) /* (h + c)) - ((SVF1 (3,(pdiff1 (f,3)),u0)) /* c))) )

proof end;

theorem :: PDIFF_5:46

for u0 being Element of REAL 3
for f1, f2 being PartFunc of (REAL 3),REAL st f1 is_hpartial_differentiable`11_in u0 & f2 is_hpartial_differentiable`11_in u0 holds
( (pdiff1 (f1,1)) + (pdiff1 (f2,1)) is_partial_differentiable_in u0,1 & partdiff (((pdiff1 (f1,1)) + (pdiff1 (f2,1))),u0,1) = (hpartdiff11 (f1,u0)) + (hpartdiff11 (f2,u0)) )

proof end;

theorem :: PDIFF_5:47

for u0 being Element of REAL 3
for f1, f2 being PartFunc of (REAL 3),REAL st f1 is_hpartial_differentiable`12_in u0 & f2 is_hpartial_differentiable`12_in u0 holds
( (pdiff1 (f1,1)) + (pdiff1 (f2,1)) is_partial_differentiable_in u0,2 & partdiff (((pdiff1 (f1,1)) + (pdiff1 (f2,1))),u0,2) = (hpartdiff12 (f1,u0)) + (hpartdiff12 (f2,u0)) )

proof end;

theorem :: PDIFF_5:48

for u0 being Element of REAL 3
for f1, f2 being PartFunc of (REAL 3),REAL st f1 is_hpartial_differentiable`13_in u0 & f2 is_hpartial_differentiable`13_in u0 holds
( (pdiff1 (f1,1)) + (pdiff1 (f2,1)) is_partial_differentiable_in u0,3 & partdiff (((pdiff1 (f1,1)) + (pdiff1 (f2,1))),u0,3) = (hpartdiff13 (f1,u0)) + (hpartdiff13 (f2,u0)) )

proof end;

theorem :: PDIFF_5:49

for u0 being Element of REAL 3
for f1, f2 being PartFunc of (REAL 3),REAL st f1 is_hpartial_differentiable`21_in u0 & f2 is_hpartial_differentiable`21_in u0 holds
( (pdiff1 (f1,2)) + (pdiff1 (f2,2)) is_partial_differentiable_in u0,1 & partdiff (((pdiff1 (f1,2)) + (pdiff1 (f2,2))),u0,1) = (hpartdiff21 (f1,u0)) + (hpartdiff21 (f2,u0)) )

proof end;

theorem :: PDIFF_5:50

for u0 being Element of REAL 3
for f1, f2 being PartFunc of (REAL 3),REAL st f1 is_hpartial_differentiable`22_in u0 & f2 is_hpartial_differentiable`22_in u0 holds
( (pdiff1 (f1,2)) + (pdiff1 (f2,2)) is_partial_differentiable_in u0,2 & partdiff (((pdiff1 (f1,2)) + (pdiff1 (f2,2))),u0,2) = (hpartdiff22 (f1,u0)) + (hpartdiff22 (f2,u0)) )

proof end;

theorem :: PDIFF_5:51

for u0 being Element of REAL 3
for f1, f2 being PartFunc of (REAL 3),REAL st f1 is_hpartial_differentiable`23_in u0 & f2 is_hpartial_differentiable`23_in u0 holds
( (pdiff1 (f1,2)) + (pdiff1 (f2,2)) is_partial_differentiable_in u0,3 & partdiff (((pdiff1 (f1,2)) + (pdiff1 (f2,2))),u0,3) = (hpartdiff23 (f1,u0)) + (hpartdiff23 (f2,u0)) )

proof end;

theorem :: PDIFF_5:52

for u0 being Element of REAL 3
for f1, f2 being PartFunc of (REAL 3),REAL st f1 is_hpartial_differentiable`11_in u0 & f2 is_hpartial_differentiable`11_in u0 holds
( (pdiff1 (f1,1)) - (pdiff1 (f2,1)) is_partial_differentiable_in u0,1 & partdiff (((pdiff1 (f1,1)) - (pdiff1 (f2,1))),u0,1) = (hpartdiff11 (f1,u0)) - (hpartdiff11 (f2,u0)) )

proof end;

theorem :: PDIFF_5:53

for u0 being Element of REAL 3
for f1, f2 being PartFunc of (REAL 3),REAL st f1 is_hpartial_differentiable`12_in u0 & f2 is_hpartial_differentiable`12_in u0 holds
( (pdiff1 (f1,1)) - (pdiff1 (f2,1)) is_partial_differentiable_in u0,2 & partdiff (((pdiff1 (f1,1)) - (pdiff1 (f2,1))),u0,2) = (hpartdiff12 (f1,u0)) - (hpartdiff12 (f2,u0)) )

proof end;

theorem :: PDIFF_5:54

for u0 being Element of REAL 3
for f1, f2 being PartFunc of (REAL 3),REAL st f1 is_hpartial_differentiable`13_in u0 & f2 is_hpartial_differentiable`13_in u0 holds
( (pdiff1 (f1,1)) - (pdiff1 (f2,1)) is_partial_differentiable_in u0,3 & partdiff (((pdiff1 (f1,1)) - (pdiff1 (f2,1))),u0,3) = (hpartdiff13 (f1,u0)) - (hpartdiff13 (f2,u0)) )

proof end;

theorem :: PDIFF_5:55

for u0 being Element of REAL 3
for f1, f2 being PartFunc of (REAL 3),REAL st f1 is_hpartial_differentiable`21_in u0 & f2 is_hpartial_differentiable`21_in u0 holds
( (pdiff1 (f1,2)) - (pdiff1 (f2,2)) is_partial_differentiable_in u0,1 & partdiff (((pdiff1 (f1,2)) - (pdiff1 (f2,2))),u0,1) = (hpartdiff21 (f1,u0)) - (hpartdiff21 (f2,u0)) )

proof end;

theorem :: PDIFF_5:56

for u0 being Element of REAL 3
for f1, f2 being PartFunc of (REAL 3),REAL st f1 is_hpartial_differentiable`22_in u0 & f2 is_hpartial_differentiable`22_in u0 holds
( (pdiff1 (f1,2)) - (pdiff1 (f2,2)) is_partial_differentiable_in u0,2 & partdiff (((pdiff1 (f1,2)) - (pdiff1 (f2,2))),u0,2) = (hpartdiff22 (f1,u0)) - (hpartdiff22 (f2,u0)) )

proof end;

theorem :: PDIFF_5:57

for u0 being Element of REAL 3
for f1, f2 being PartFunc of (REAL 3),REAL st f1 is_hpartial_differentiable`23_in u0 & f2 is_hpartial_differentiable`23_in u0 holds
( (pdiff1 (f1,2)) - (pdiff1 (f2,2)) is_partial_differentiable_in u0,3 & partdiff (((pdiff1 (f1,2)) - (pdiff1 (f2,2))),u0,3) = (hpartdiff23 (f1,u0)) - (hpartdiff23 (f2,u0)) )

proof end;

theorem :: PDIFF_5:58

for r being Real
for u0 being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st f is_hpartial_differentiable`11_in u0 holds
( r (#) (pdiff1 (f,1)) is_partial_differentiable_in u0,1 & partdiff ((r (#) (pdiff1 (f,1))),u0,1) = r * (hpartdiff11 (f,u0)) )

proof end;

theorem :: PDIFF_5:59

for r being Real
for u0 being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st f is_hpartial_differentiable`12_in u0 holds
( r (#) (pdiff1 (f,1)) is_partial_differentiable_in u0,2 & partdiff ((r (#) (pdiff1 (f,1))),u0,2) = r * (hpartdiff12 (f,u0)) )

proof end;

theorem :: PDIFF_5:60

for r being Real
for u0 being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st f is_hpartial_differentiable`13_in u0 holds
( r (#) (pdiff1 (f,1)) is_partial_differentiable_in u0,3 & partdiff ((r (#) (pdiff1 (f,1))),u0,3) = r * (hpartdiff13 (f,u0)) )

proof end;

theorem :: PDIFF_5:61

for r being Real
for u0 being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st f is_hpartial_differentiable`21_in u0 holds
( r (#) (pdiff1 (f,2)) is_partial_differentiable_in u0,1 & partdiff ((r (#) (pdiff1 (f,2))),u0,1) = r * (hpartdiff21 (f,u0)) )

proof end;

theorem :: PDIFF_5:62

for r being Real
for u0 being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st f is_hpartial_differentiable`22_in u0 holds
( r (#) (pdiff1 (f,2)) is_partial_differentiable_in u0,2 & partdiff ((r (#) (pdiff1 (f,2))),u0,2) = r * (hpartdiff22 (f,u0)) )

proof end;

theorem :: PDIFF_5:63

for r being Real
for u0 being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st f is_hpartial_differentiable`23_in u0 holds
( r (#) (pdiff1 (f,2)) is_partial_differentiable_in u0,3 & partdiff ((r (#) (pdiff1 (f,2))),u0,3) = r * (hpartdiff23 (f,u0)) )

proof end;

theorem :: PDIFF_5:64

for r being Real
for u0 being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st f is_hpartial_differentiable`31_in u0 holds
( r (#) (pdiff1 (f,3)) is_partial_differentiable_in u0,1 & partdiff ((r (#) (pdiff1 (f,3))),u0,1) = r * (hpartdiff31 (f,u0)) )

proof end;

theorem :: PDIFF_5:65

for r being Real
for u0 being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st f is_hpartial_differentiable`32_in u0 holds
( r (#) (pdiff1 (f,3)) is_partial_differentiable_in u0,2 & partdiff ((r (#) (pdiff1 (f,3))),u0,2) = r * (hpartdiff32 (f,u0)) )

proof end;

theorem :: PDIFF_5:66

for r being Real
for u0 being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st f is_hpartial_differentiable`33_in u0 holds
( r (#) (pdiff1 (f,3)) is_partial_differentiable_in u0,3 & partdiff ((r (#) (pdiff1 (f,3))),u0,3) = r * (hpartdiff33 (f,u0)) )

proof end;

theorem :: PDIFF_5:67

for u0 being Element of REAL 3
for f1, f2 being PartFunc of (REAL 3),REAL st f1 is_hpartial_differentiable`11_in u0 & f2 is_hpartial_differentiable`11_in u0 holds
(pdiff1 (f1,1)) (#) (pdiff1 (f2,1)) is_partial_differentiable_in u0,1

proof end;

theorem :: PDIFF_5:68

for u0 being Element of REAL 3
for f1, f2 being PartFunc of (REAL 3),REAL st f1 is_hpartial_differentiable`12_in u0 & f2 is_hpartial_differentiable`12_in u0 holds
(pdiff1 (f1,1)) (#) (pdiff1 (f2,1)) is_partial_differentiable_in u0,2

proof end;

theorem :: PDIFF_5:69

for u0 being Element of REAL 3
for f1, f2 being PartFunc of (REAL 3),REAL st f1 is_hpartial_differentiable`13_in u0 & f2 is_hpartial_differentiable`13_in u0 holds
(pdiff1 (f1,1)) (#) (pdiff1 (f2,1)) is_partial_differentiable_in u0,3

proof end;

theorem :: PDIFF_5:70

for u0 being Element of REAL 3
for f1, f2 being PartFunc of (REAL 3),REAL st f1 is_hpartial_differentiable`21_in u0 & f2 is_hpartial_differentiable`21_in u0 holds
(pdiff1 (f1,2)) (#) (pdiff1 (f2,2)) is_partial_differentiable_in u0,1

proof end;

theorem :: PDIFF_5:71

for u0 being Element of REAL 3
for f1, f2 being PartFunc of (REAL 3),REAL st f1 is_hpartial_differentiable`22_in u0 & f2 is_hpartial_differentiable`22_in u0 holds
(pdiff1 (f1,2)) (#) (pdiff1 (f2,2)) is_partial_differentiable_in u0,2

proof end;

theorem :: PDIFF_5:72

for u0 being Element of REAL 3
for f1, f2 being PartFunc of (REAL 3),REAL st f1 is_hpartial_differentiable`23_in u0 & f2 is_hpartial_differentiable`23_in u0 holds
(pdiff1 (f1,2)) (#) (pdiff1 (f2,2)) is_partial_differentiable_in u0,3

proof end;

theorem :: PDIFF_5:73

for u0 being Element of REAL 3
for f1, f2 being PartFunc of (REAL 3),REAL st f1 is_hpartial_differentiable`31_in u0 & f2 is_hpartial_differentiable`31_in u0 holds
(pdiff1 (f1,3)) (#) (pdiff1 (f2,3)) is_partial_differentiable_in u0,1

proof end;

theorem :: PDIFF_5:74

for u0 being Element of REAL 3
for f1, f2 being PartFunc of (REAL 3),REAL st f1 is_hpartial_differentiable`32_in u0 & f2 is_hpartial_differentiable`32_in u0 holds
(pdiff1 (f1,3)) (#) (pdiff1 (f2,3)) is_partial_differentiable_in u0,2

proof end;

theorem :: PDIFF_5:75

for u0 being Element of REAL 3
for f1, f2 being PartFunc of (REAL 3),REAL st f1 is_hpartial_differentiable`33_in u0 & f2 is_hpartial_differentiable`33_in u0 holds
(pdiff1 (f1,3)) (#) (pdiff1 (f2,3)) is_partial_differentiable_in u0,3

proof end;

theorem :: PDIFF_5:76

for f being PartFunc of (REAL 3),REAL
for u0 being Element of REAL 3 st f is_hpartial_differentiable`11_in u0 holds
SVF1 (1,(pdiff1 (f,1)),u0) is_continuous_in (proj (1,3)) . u0 by Th19, PDIFF_4:31;

theorem :: PDIFF_5:77

for f being PartFunc of (REAL 3),REAL
for u0 being Element of REAL 3 st f is_hpartial_differentiable`12_in u0 holds
SVF1 (2,(pdiff1 (f,1)),u0) is_continuous_in (proj (2,3)) . u0 by Th20, PDIFF_4:32;

theorem :: PDIFF_5:78

for f being PartFunc of (REAL 3),REAL
for u0 being Element of REAL 3 st f is_hpartial_differentiable`13_in u0 holds
SVF1 (3,(pdiff1 (f,1)),u0) is_continuous_in (proj (3,3)) . u0 by Th21, PDIFF_4:33;

theorem :: PDIFF_5:79

for f being PartFunc of (REAL 3),REAL
for u0 being Element of REAL 3 st f is_hpartial_differentiable`21_in u0 holds
SVF1 (1,(pdiff1 (f,2)),u0) is_continuous_in (proj (1,3)) . u0 by Th22, PDIFF_4:31;

theorem :: PDIFF_5:80

for f being PartFunc of (REAL 3),REAL
for u0 being Element of REAL 3 st f is_hpartial_differentiable`22_in u0 holds
SVF1 (2,(pdiff1 (f,2)),u0) is_continuous_in (proj (2,3)) . u0 by Th23, PDIFF_4:32;

theorem :: PDIFF_5:81

for f being PartFunc of (REAL 3),REAL
for u0 being Element of REAL 3 st f is_hpartial_differentiable`23_in u0 holds
SVF1 (3,(pdiff1 (f,2)),u0) is_continuous_in (proj (3,3)) . u0 by Th24, PDIFF_4:33;

theorem :: PDIFF_5:82

for f being PartFunc of (REAL 3),REAL
for u0 being Element of REAL 3 st f is_hpartial_differentiable`31_in u0 holds
SVF1 (1,(pdiff1 (f,3)),u0) is_continuous_in (proj (1,3)) . u0 by Th25, PDIFF_4:31;

theorem :: PDIFF_5:83

for f being PartFunc of (REAL 3),REAL
for u0 being Element of REAL 3 st f is_hpartial_differentiable`32_in u0 holds
SVF1 (2,(pdiff1 (f,3)),u0) is_continuous_in (proj (2,3)) . u0 by Th26, PDIFF_4:32;

theorem :: PDIFF_5:84

for f being PartFunc of (REAL 3),REAL
for u0 being Element of REAL 3 st f is_hpartial_differentiable`33_in u0 holds
SVF1 (3,(pdiff1 (f,3)),u0) is_continuous_in (proj (3,3)) . u0 by Th27, PDIFF_4:33;