:: Second-order Partial Differentiation of Real Ternary Functions
:: by Takao Inou\'e
::
:: 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_differentiable11_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_differentiable12_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_differentiable13_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_differentiable21_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_differentiable22_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_differentiable23_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_differentiable31_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_differentiable32_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_differentiable33_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_differentiable11_in PDIFF_5:def 1 :
for f being PartFunc of (REAL 3),REAL
for u being Element of REAL 3 holds
( f is_hpartial_differentiable11_in u iff 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)) ) ) );

:: deftheorem defines is_hpartial_differentiable12_in PDIFF_5:def 2 :
for f being PartFunc of (REAL 3),REAL
for u being Element of REAL 3 holds
( f is_hpartial_differentiable12_in u iff 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)) ) ) );

:: deftheorem defines is_hpartial_differentiable13_in PDIFF_5:def 3 :
for f being PartFunc of (REAL 3),REAL
for u being Element of REAL 3 holds
( f is_hpartial_differentiable13_in u iff 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)) ) ) );

:: deftheorem defines is_hpartial_differentiable21_in PDIFF_5:def 4 :
for f being PartFunc of (REAL 3),REAL
for u being Element of REAL 3 holds
( f is_hpartial_differentiable21_in u iff 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)) ) ) );

:: deftheorem defines is_hpartial_differentiable22_in PDIFF_5:def 5 :
for f being PartFunc of (REAL 3),REAL
for u being Element of REAL 3 holds
( f is_hpartial_differentiable22_in u iff 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)) ) ) );

:: deftheorem defines is_hpartial_differentiable23_in PDIFF_5:def 6 :
for f being PartFunc of (REAL 3),REAL
for u being Element of REAL 3 holds
( f is_hpartial_differentiable23_in u iff 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)) ) ) );

:: deftheorem defines is_hpartial_differentiable31_in PDIFF_5:def 7 :
for f being PartFunc of (REAL 3),REAL
for u being Element of REAL 3 holds
( f is_hpartial_differentiable31_in u iff 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)) ) ) );

:: deftheorem defines is_hpartial_differentiable32_in PDIFF_5:def 8 :
for f being PartFunc of (REAL 3),REAL
for u being Element of REAL 3 holds
( f is_hpartial_differentiable32_in u iff 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)) ) ) );

:: deftheorem defines is_hpartial_differentiable33_in PDIFF_5:def 9 :
for f being PartFunc of (REAL 3),REAL
for u being Element of REAL 3 holds
( f is_hpartial_differentiable33_in u iff 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)) ) ) );

definition
let f be PartFunc of (REAL 3),REAL;
let u be Element of REAL 3;
assume A1: f is_hpartial_differentiable11_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)) ) ) ) );
existence
ex b1, 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
( b1 = 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;
uniqueness
for b1, b2 being Real st 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
( b1 = 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)) ) ) ) ) & 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
( b2 = 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)) ) ) ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def10 defines hpartdiff11 PDIFF_5:def 10 :
for f being PartFunc of (REAL 3),REAL
for u being Element of REAL 3 st f is_hpartial_differentiable11_in u holds
for b3 being Real holds
( b3 = hpartdiff11 (f,u) iff 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
( b3 = 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)) ) ) ) ) );

definition
let f be PartFunc of (REAL 3),REAL;
let u be Element of REAL 3;
assume A1: f is_hpartial_differentiable12_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)) ) ) ) );
existence
ex b1, 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
( b1 = 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;
uniqueness
for b1, b2 being Real st 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
( b1 = 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)) ) ) ) ) & 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
( b2 = 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)) ) ) ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def11 defines hpartdiff12 PDIFF_5:def 11 :
for f being PartFunc of (REAL 3),REAL
for u being Element of REAL 3 st f is_hpartial_differentiable12_in u holds
for b3 being Real holds
( b3 = hpartdiff12 (f,u) iff 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
( b3 = 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)) ) ) ) ) );

definition
let f be PartFunc of (REAL 3),REAL;
let u be Element of REAL 3;
assume A1: f is_hpartial_differentiable13_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)) ) ) ) );
existence
ex b1, 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
( b1 = 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;
uniqueness
for b1, b2 being Real st 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
( b1 = 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)) ) ) ) ) & 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
( b2 = 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)) ) ) ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def12 defines hpartdiff13 PDIFF_5:def 12 :
for f being PartFunc of (REAL 3),REAL
for u being Element of REAL 3 st f is_hpartial_differentiable13_in u holds
for b3 being Real holds
( b3 = hpartdiff13 (f,u) iff 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
( b3 = 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)) ) ) ) ) );

definition
let f be PartFunc of (REAL 3),REAL;
let u be Element of REAL 3;
assume A1: f is_hpartial_differentiable21_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)) ) ) ) );
existence
ex b1, 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
( b1 = 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;
uniqueness
for b1, b2 being Real st 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
( b1 = 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)) ) ) ) ) & 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
( b2 = 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)) ) ) ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def13 defines hpartdiff21 PDIFF_5:def 13 :
for f being PartFunc of (REAL 3),REAL
for u being Element of REAL 3 st f is_hpartial_differentiable21_in u holds
for b3 being Real holds
( b3 = hpartdiff21 (f,u) iff 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
( b3 = 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)) ) ) ) ) );

definition
let f be PartFunc of (REAL 3),REAL;
let u be Element of REAL 3;
assume A1: f is_hpartial_differentiable22_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)) ) ) ) );
existence
ex b1, 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
( b1 = 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;
uniqueness
for b1, b2 being Real st 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
( b1 = 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)) ) ) ) ) & 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
( b2 = 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)) ) ) ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def14 defines hpartdiff22 PDIFF_5:def 14 :
for f being PartFunc of (REAL 3),REAL
for u being Element of REAL 3 st f is_hpartial_differentiable22_in u holds
for b3 being Real holds
( b3 = hpartdiff22 (f,u) iff 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
( b3 = 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)) ) ) ) ) );

definition
let f be PartFunc of (REAL 3),REAL;
let u be Element of REAL 3;
assume A1: f is_hpartial_differentiable23_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)) ) ) ) );
existence
ex b1, 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
( b1 = 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;
uniqueness
for b1, b2 being Real st 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
( b1 = 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)) ) ) ) ) & 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
( b2 = 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)) ) ) ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def15 defines hpartdiff23 PDIFF_5:def 15 :
for f being PartFunc of (REAL 3),REAL
for u being Element of REAL 3 st f is_hpartial_differentiable23_in u holds
for b3 being Real holds
( b3 = hpartdiff23 (f,u) iff 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
( b3 = 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)) ) ) ) ) );

definition
let f be PartFunc of (REAL 3),REAL;
let u be Element of REAL 3;
assume A1: f is_hpartial_differentiable31_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)) ) ) ) );
existence
ex b1, 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
( b1 = 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;
uniqueness
for b1, b2 being Real st 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
( b1 = 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)) ) ) ) ) & 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
( b2 = 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)) ) ) ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def16 defines hpartdiff31 PDIFF_5:def 16 :
for f being PartFunc of (REAL 3),REAL
for u being Element of REAL 3 st f is_hpartial_differentiable31_in u holds
for b3 being Real holds
( b3 = hpartdiff31 (f,u) iff 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
( b3 = 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)) ) ) ) ) );

definition
let f be PartFunc of (REAL 3),REAL;
let u be Element of REAL 3;
assume A1: f is_hpartial_differentiable32_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)) ) ) ) );
existence
ex b1, 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
( b1 = 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;
uniqueness
for b1, b2 being Real st 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
( b1 = 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)) ) ) ) ) & 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
( b2 = 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)) ) ) ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def17 defines hpartdiff32 PDIFF_5:def 17 :
for f being PartFunc of (REAL 3),REAL
for u being Element of REAL 3 st f is_hpartial_differentiable32_in u holds
for b3 being Real holds
( b3 = hpartdiff32 (f,u) iff 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
( b3 = 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)) ) ) ) ) );

definition
let f be PartFunc of (REAL 3),REAL;
let u be Element of REAL 3;
assume A1: f is_hpartial_differentiable33_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)) ) ) ) );
existence
ex b1, 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
( b1 = 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;
uniqueness
for b1, b2 being Real st 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
( b1 = 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)) ) ) ) ) & 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
( b2 = 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)) ) ) ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def18 defines hpartdiff33 PDIFF_5:def 18 :
for f being PartFunc of (REAL 3),REAL
for u being Element of REAL 3 st f is_hpartial_differentiable33_in u holds
for b3 being Real holds
( b3 = hpartdiff33 (f,u) iff 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
( b3 = 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)) ) ) ) ) );

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_differentiable11_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_differentiable12_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_differentiable13_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_differentiable21_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_differentiable22_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_differentiable23_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_differentiable31_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_differentiable32_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_differentiable33_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_differentiable11_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_differentiable12_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_differentiable13_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_differentiable21_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_differentiable22_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_differentiable23_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_differentiable31_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_differentiable32_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_differentiable33_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_differentiable11_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_differentiable11_in u ) );
pred f is_hpartial_differentiable12_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_differentiable12_in u ) );
pred f is_hpartial_differentiable13_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_differentiable13_in u ) );
pred f is_hpartial_differentiable21_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_differentiable21_in u ) );
pred f is_hpartial_differentiable22_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_differentiable22_in u ) );
pred f is_hpartial_differentiable23_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_differentiable23_in u ) );
pred f is_hpartial_differentiable31_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_differentiable31_in u ) );
pred f is_hpartial_differentiable32_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_differentiable32_in u ) );
pred f is_hpartial_differentiable33_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_differentiable33_in u ) );
end;

:: deftheorem defines is_hpartial_differentiable11_on PDIFF_5:def 19 :
for f being PartFunc of (REAL 3),REAL
for D being set holds
( f is_hpartial_differentiable11_on D iff ( D c= dom f & ( for u being Element of REAL 3 st u in D holds
f | D is_hpartial_differentiable11_in u ) ) );

:: deftheorem defines is_hpartial_differentiable12_on PDIFF_5:def 20 :
for f being PartFunc of (REAL 3),REAL
for D being set holds
( f is_hpartial_differentiable12_on D iff ( D c= dom f & ( for u being Element of REAL 3 st u in D holds
f | D is_hpartial_differentiable12_in u ) ) );

:: deftheorem defines is_hpartial_differentiable13_on PDIFF_5:def 21 :
for f being PartFunc of (REAL 3),REAL
for D being set holds
( f is_hpartial_differentiable13_on D iff ( D c= dom f & ( for u being Element of REAL 3 st u in D holds
f | D is_hpartial_differentiable13_in u ) ) );

:: deftheorem defines is_hpartial_differentiable21_on PDIFF_5:def 22 :
for f being PartFunc of (REAL 3),REAL
for D being set holds
( f is_hpartial_differentiable21_on D iff ( D c= dom f & ( for u being Element of REAL 3 st u in D holds
f | D is_hpartial_differentiable21_in u ) ) );

:: deftheorem defines is_hpartial_differentiable22_on PDIFF_5:def 23 :
for f being PartFunc of (REAL 3),REAL
for D being set holds
( f is_hpartial_differentiable22_on D iff ( D c= dom f & ( for u being Element of REAL 3 st u in D holds
f | D is_hpartial_differentiable22_in u ) ) );

:: deftheorem defines is_hpartial_differentiable23_on PDIFF_5:def 24 :
for f being PartFunc of (REAL 3),REAL
for D being set holds
( f is_hpartial_differentiable23_on D iff ( D c= dom f & ( for u being Element of REAL 3 st u in D holds
f | D is_hpartial_differentiable23_in u ) ) );

:: deftheorem defines is_hpartial_differentiable31_on PDIFF_5:def 25 :
for f being PartFunc of (REAL 3),REAL
for D being set holds
( f is_hpartial_differentiable31_on D iff ( D c= dom f & ( for u being Element of REAL 3 st u in D holds
f | D is_hpartial_differentiable31_in u ) ) );

:: deftheorem defines is_hpartial_differentiable32_on PDIFF_5:def 26 :
for f being PartFunc of (REAL 3),REAL
for D being set holds
( f is_hpartial_differentiable32_on D iff ( D c= dom f & ( for u being Element of REAL 3 st u in D holds
f | D is_hpartial_differentiable32_in u ) ) );

:: deftheorem defines is_hpartial_differentiable33_on PDIFF_5:def 27 :
for f being PartFunc of (REAL 3),REAL
for D being set holds
( f is_hpartial_differentiable33_on D iff ( D c= dom f & ( for u being Element of REAL 3 st u in D holds
f | D is_hpartial_differentiable33_in u ) ) );

definition
let f be PartFunc of (REAL 3),REAL;
let D be set ;
assume A1: f is_hpartial_differentiable11_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) ) );
existence
ex b1 being PartFunc of (REAL 3),REAL st
( dom b1 = D & ( for u being Element of REAL 3 st u in D holds
b1 . u = hpartdiff11 (f,u) ) )
proof end;
uniqueness
for b1, b2 being PartFunc of (REAL 3),REAL st dom b1 = D & ( for u being Element of REAL 3 st u in D holds
b1 . u = hpartdiff11 (f,u) ) & dom b2 = D & ( for u being Element of REAL 3 st u in D holds
b2 . u = hpartdiff11 (f,u) ) holds
b1 = b2
proof end;
end;

:: deftheorem defines hpartial11| PDIFF_5:def 28 :
for f being PartFunc of (REAL 3),REAL
for D being set st f is_hpartial_differentiable11_on D holds
for b3 being PartFunc of (REAL 3),REAL holds
( b3 = f hpartial11| D iff ( dom b3 = D & ( for u being Element of REAL 3 st u in D holds
b3 . u = hpartdiff11 (f,u) ) ) );

definition
let f be PartFunc of (REAL 3),REAL;
let D be set ;
assume A1: f is_hpartial_differentiable12_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) ) );
existence
ex b1 being PartFunc of (REAL 3),REAL st
( dom b1 = D & ( for u being Element of REAL 3 st u in D holds
b1 . u = hpartdiff12 (f,u) ) )
proof end;
uniqueness
for b1, b2 being PartFunc of (REAL 3),REAL st dom b1 = D & ( for u being Element of REAL 3 st u in D holds
b1 . u = hpartdiff12 (f,u) ) & dom b2 = D & ( for u being Element of REAL 3 st u in D holds
b2 . u = hpartdiff12 (f,u) ) holds
b1 = b2
proof end;
end;

:: deftheorem defines hpartial12| PDIFF_5:def 29 :
for f being PartFunc of (REAL 3),REAL
for D being set st f is_hpartial_differentiable12_on D holds
for b3 being PartFunc of (REAL 3),REAL holds
( b3 = f hpartial12| D iff ( dom b3 = D & ( for u being Element of REAL 3 st u in D holds
b3 . u = hpartdiff12 (f,u) ) ) );

definition
let f be PartFunc of (REAL 3),REAL;
let D be set ;
assume A1: f is_hpartial_differentiable13_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) ) );
existence
ex b1 being PartFunc of (REAL 3),REAL st
( dom b1 = D & ( for u being Element of REAL 3 st u in D holds
b1 . u = hpartdiff13 (f,u) ) )
proof end;
uniqueness
for b1, b2 being PartFunc of (REAL 3),REAL st dom b1 = D & ( for u being Element of REAL 3 st u in D holds
b1 . u = hpartdiff13 (f,u) ) & dom b2 = D & ( for u being Element of REAL 3 st u in D holds
b2 . u = hpartdiff13 (f,u) ) holds
b1 = b2
proof end;
end;

:: deftheorem defines hpartial13| PDIFF_5:def 30 :
for f being PartFunc of (REAL 3),REAL
for D being set st f is_hpartial_differentiable13_on D holds
for b3 being PartFunc of (REAL 3),REAL holds
( b3 = f hpartial13| D iff ( dom b3 = D & ( for u being Element of REAL 3 st u in D holds
b3 . u = hpartdiff13 (f,u) ) ) );

definition
let f be PartFunc of (REAL 3),REAL;
let D be set ;
assume A1: f is_hpartial_differentiable21_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) ) );
existence
ex b1 being PartFunc of (REAL 3),REAL st
( dom b1 = D & ( for u being Element of REAL 3 st u in D holds
b1 . u = hpartdiff21 (f,u) ) )
proof end;
uniqueness
for b1, b2 being PartFunc of (REAL 3),REAL st dom b1 = D & ( for u being Element of REAL 3 st u in D holds
b1 . u = hpartdiff21 (f,u) ) & dom b2 = D & ( for u being Element of REAL 3 st u in D holds
b2 . u = hpartdiff21 (f,u) ) holds
b1 = b2
proof end;
end;

:: deftheorem defines hpartial21| PDIFF_5:def 31 :
for f being PartFunc of (REAL 3),REAL
for D being set st f is_hpartial_differentiable21_on D holds
for b3 being PartFunc of (REAL 3),REAL holds
( b3 = f hpartial21| D iff ( dom b3 = D & ( for u being Element of REAL 3 st u in D holds
b3 . u = hpartdiff21 (f,u) ) ) );

definition
let f be PartFunc of (REAL 3),REAL;
let D be set ;
assume A1: f is_hpartial_differentiable22_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) ) );
existence
ex b1 being PartFunc of (REAL 3),REAL st
( dom b1 = D & ( for u being Element of REAL 3 st u in D holds
b1 . u = hpartdiff22 (f,u) ) )
proof end;
uniqueness
for b1, b2 being PartFunc of (REAL 3),REAL st dom b1 = D & ( for u being Element of REAL 3 st u in D holds
b1 . u = hpartdiff22 (f,u) ) & dom b2 = D & ( for u being Element of REAL 3 st u in D holds
b2 . u = hpartdiff22 (f,u) ) holds
b1 = b2
proof end;
end;

:: deftheorem defines hpartial22| PDIFF_5:def 32 :
for f being PartFunc of (REAL 3),REAL
for D being set st f is_hpartial_differentiable22_on D holds
for b3 being PartFunc of (REAL 3),REAL holds
( b3 = f hpartial22| D iff ( dom b3 = D & ( for u being Element of REAL 3 st u in D holds
b3 . u = hpartdiff22 (f,u) ) ) );

definition
let f be PartFunc of (REAL 3),REAL;
let D be set ;
assume A1: f is_hpartial_differentiable23_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) ) );
existence
ex b1 being PartFunc of (REAL 3),REAL st
( dom b1 = D & ( for u being Element of REAL 3 st u in D holds
b1 . u = hpartdiff23 (f,u) ) )
proof end;
uniqueness
for b1, b2 being PartFunc of (REAL 3),REAL st dom b1 = D & ( for u being Element of REAL 3 st u in D holds
b1 . u = hpartdiff23 (f,u) ) & dom b2 = D & ( for u being Element of REAL 3 st u in D holds
b2 . u = hpartdiff23 (f,u) ) holds
b1 = b2
proof end;
end;

:: deftheorem defines hpartial23| PDIFF_5:def 33 :
for f being PartFunc of (REAL 3),REAL
for D being set st f is_hpartial_differentiable23_on D holds
for b3 being PartFunc of (REAL 3),REAL holds
( b3 = f hpartial23| D iff ( dom b3 = D & ( for u being Element of REAL 3 st u in D holds
b3 . u = hpartdiff23 (f,u) ) ) );

definition
let f be PartFunc of (REAL 3),REAL;
let D be set ;
assume A1: f is_hpartial_differentiable31_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) ) );
existence
ex b1 being PartFunc of (REAL 3),REAL st
( dom b1 = D & ( for u being Element of REAL 3 st u in D holds
b1 . u = hpartdiff31 (f,u) ) )
proof end;
uniqueness
for b1, b2 being PartFunc of (REAL 3),REAL st dom b1 = D & ( for u being Element of REAL 3 st u in D holds
b1 . u = hpartdiff31 (f,u) ) & dom b2 = D & ( for u being Element of REAL 3 st u in D holds
b2 . u = hpartdiff31 (f,u) ) holds
b1 = b2
proof end;
end;

:: deftheorem defines hpartial31| PDIFF_5:def 34 :
for f being PartFunc of (REAL 3),REAL
for D being set st f is_hpartial_differentiable31_on D holds
for b3 being PartFunc of (REAL 3),REAL holds
( b3 = f hpartial31| D iff ( dom b3 = D & ( for u being Element of REAL 3 st u in D holds
b3 . u = hpartdiff31 (f,u) ) ) );

definition
let f be PartFunc of (REAL 3),REAL;
let D be set ;
assume A1: f is_hpartial_differentiable32_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) ) );
existence
ex b1 being PartFunc of (REAL 3),REAL st
( dom b1 = D & ( for u being Element of REAL 3 st u in D holds
b1 . u = hpartdiff32 (f,u) ) )
proof end;
uniqueness
for b1, b2 being PartFunc of (REAL 3),REAL st dom b1 = D & ( for u being Element of REAL 3 st u in D holds
b1 . u = hpartdiff32 (f,u) ) & dom b2 = D & ( for u being Element of REAL 3 st u in D holds
b2 . u = hpartdiff32 (f,u) ) holds
b1 = b2
proof end;
end;

:: deftheorem defines hpartial32| PDIFF_5:def 35 :
for f being PartFunc of (REAL 3),REAL
for D being set st f is_hpartial_differentiable32_on D holds
for b3 being PartFunc of (REAL 3),REAL holds
( b3 = f hpartial32| D iff ( dom b3 = D & ( for u being Element of REAL 3 st u in D holds
b3 . u = hpartdiff32 (f,u) ) ) );

definition
let f be PartFunc of (REAL 3),REAL;
let D be set ;
assume A1: f is_hpartial_differentiable33_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) ) );
existence
ex b1 being PartFunc of (REAL 3),REAL st
( dom b1 = D & ( for u being Element of REAL 3 st u in D holds
b1 . u = hpartdiff33 (f,u) ) )
proof end;
uniqueness
for b1, b2 being PartFunc of (REAL 3),REAL st dom b1 = D & ( for u being Element of REAL 3 st u in D holds
b1 . u = hpartdiff33 (f,u) ) & dom b2 = D & ( for u being Element of REAL 3 st u in D holds
b2 . u = hpartdiff33 (f,u) ) holds
b1 = b2
proof end;
end;

:: deftheorem defines hpartial33| PDIFF_5:def 36 :
for f being PartFunc of (REAL 3),REAL
for D being set st f is_hpartial_differentiable33_on D holds
for b3 being PartFunc of (REAL 3),REAL holds
( b3 = f hpartial33| D iff ( dom b3 = D & ( for u being Element of REAL 3 st u in D holds
b3 . u = hpartdiff33 (f,u) ) ) );

theorem Th19: :: PDIFF_5:19
for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL holds
( f is_hpartial_differentiable11_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_differentiable12_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_differentiable13_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_differentiable21_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_differentiable22_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_differentiable23_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_differentiable31_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_differentiable32_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_differentiable33_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_differentiable11_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_differentiable12_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_differentiable13_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_differentiable21_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_differentiable22_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_differentiable23_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_differentiable31_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_differentiable32_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_differentiable33_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_differentiable11_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_differentiable12_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_differentiable13_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_differentiable21_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_differentiable22_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_differentiable23_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_differentiable31_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_differentiable32_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_differentiable33_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_differentiable11_in u0 & f2 is_hpartial_differentiable11_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_differentiable12_in u0 & f2 is_hpartial_differentiable12_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_differentiable13_in u0 & f2 is_hpartial_differentiable13_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_differentiable21_in u0 & f2 is_hpartial_differentiable21_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_differentiable22_in u0 & f2 is_hpartial_differentiable22_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_differentiable23_in u0 & f2 is_hpartial_differentiable23_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_differentiable11_in u0 & f2 is_hpartial_differentiable11_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_differentiable12_in u0 & f2 is_hpartial_differentiable12_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_differentiable13_in u0 & f2 is_hpartial_differentiable13_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_differentiable21_in u0 & f2 is_hpartial_differentiable21_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_differentiable22_in u0 & f2 is_hpartial_differentiable22_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_differentiable23_in u0 & f2 is_hpartial_differentiable23_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_differentiable11_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_differentiable12_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_differentiable13_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_differentiable21_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_differentiable22_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_differentiable23_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_differentiable31_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_differentiable32_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_differentiable33_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_differentiable11_in u0 & f2 is_hpartial_differentiable11_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_differentiable12_in u0 & f2 is_hpartial_differentiable12_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_differentiable13_in u0 & f2 is_hpartial_differentiable13_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_differentiable21_in u0 & f2 is_hpartial_differentiable21_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_differentiable22_in u0 & f2 is_hpartial_differentiable22_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_differentiable23_in u0 & f2 is_hpartial_differentiable23_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_differentiable31_in u0 & f2 is_hpartial_differentiable31_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_differentiable32_in u0 & f2 is_hpartial_differentiable32_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_differentiable33_in u0 & f2 is_hpartial_differentiable33_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_differentiable11_in u0 holds
SVF1 (1,(pdiff1 (f,1)),u0) is_continuous_in (proj (1,3)) . u0 by ;

theorem :: PDIFF_5:77
for f being PartFunc of (REAL 3),REAL
for u0 being Element of REAL 3 st f is_hpartial_differentiable12_in u0 holds
SVF1 (2,(pdiff1 (f,1)),u0) is_continuous_in (proj (2,3)) . u0 by ;

theorem :: PDIFF_5:78
for f being PartFunc of (REAL 3),REAL
for u0 being Element of REAL 3 st f is_hpartial_differentiable13_in u0 holds
SVF1 (3,(pdiff1 (f,1)),u0) is_continuous_in (proj (3,3)) . u0 by ;

theorem :: PDIFF_5:79
for f being PartFunc of (REAL 3),REAL
for u0 being Element of REAL 3 st f is_hpartial_differentiable21_in u0 holds
SVF1 (1,(pdiff1 (f,2)),u0) is_continuous_in (proj (1,3)) . u0 by ;

theorem :: PDIFF_5:80
for f being PartFunc of (REAL 3),REAL
for u0 being Element of REAL 3 st f is_hpartial_differentiable22_in u0 holds
SVF1 (2,(pdiff1 (f,2)),u0) is_continuous_in (proj (2,3)) . u0 by ;

theorem :: PDIFF_5:81
for f being PartFunc of (REAL 3),REAL
for u0 being Element of REAL 3 st f is_hpartial_differentiable23_in u0 holds
SVF1 (3,(pdiff1 (f,2)),u0) is_continuous_in (proj (3,3)) . u0 by ;

theorem :: PDIFF_5:82
for f being PartFunc of (REAL 3),REAL
for u0 being Element of REAL 3 st f is_hpartial_differentiable31_in u0 holds
SVF1 (1,(pdiff1 (f,3)),u0) is_continuous_in (proj (1,3)) . u0 by ;

theorem :: PDIFF_5:83
for f being PartFunc of (REAL 3),REAL
for u0 being Element of REAL 3 st f is_hpartial_differentiable32_in u0 holds
SVF1 (2,(pdiff1 (f,3)),u0) is_continuous_in (proj (2,3)) . u0 by ;

theorem :: PDIFF_5:84
for f being PartFunc of (REAL 3),REAL
for u0 being Element of REAL 3 st f is_hpartial_differentiable33_in u0 holds
SVF1 (3,(pdiff1 (f,3)),u0) is_continuous_in (proj (3,3)) . u0 by ;