:: PDIFF_5 semantic presentation
begin
definition
let f be PartFunc of (REAL 3),REAL;
let u be Element of REAL 3;
predf is_hpartial_differentiable`11_in u means :Def1: :: 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)) ) );
predf is_hpartial_differentiable`12_in u means :Def2: :: 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)) ) );
predf is_hpartial_differentiable`13_in u means :Def3: :: 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)) ) );
predf is_hpartial_differentiable`21_in u means :Def4: :: 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)) ) );
predf is_hpartial_differentiable`22_in u means :Def5: :: 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)) ) );
predf is_hpartial_differentiable`23_in u means :Def6: :: 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)) ) );
predf is_hpartial_differentiable`31_in u means :Def7: :: 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)) ) );
predf is_hpartial_differentiable`32_in u means :Def8: :: 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)) ) );
predf is_hpartial_differentiable`33_in u means :Def9: :: 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 Def1 defines is_hpartial_differentiable`11_in PDIFF_5:def_1_:_
for f being PartFunc of (REAL 3),REAL
for u being Element of REAL 3 holds
( f is_hpartial_differentiable`11_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 Def2 defines is_hpartial_differentiable`12_in PDIFF_5:def_2_:_
for f being PartFunc of (REAL 3),REAL
for u being Element of REAL 3 holds
( f is_hpartial_differentiable`12_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 Def3 defines is_hpartial_differentiable`13_in PDIFF_5:def_3_:_
for f being PartFunc of (REAL 3),REAL
for u being Element of REAL 3 holds
( f is_hpartial_differentiable`13_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 Def4 defines is_hpartial_differentiable`21_in PDIFF_5:def_4_:_
for f being PartFunc of (REAL 3),REAL
for u being Element of REAL 3 holds
( f is_hpartial_differentiable`21_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 Def5 defines is_hpartial_differentiable`22_in PDIFF_5:def_5_:_
for f being PartFunc of (REAL 3),REAL
for u being Element of REAL 3 holds
( f is_hpartial_differentiable`22_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 Def6 defines is_hpartial_differentiable`23_in PDIFF_5:def_6_:_
for f being PartFunc of (REAL 3),REAL
for u being Element of REAL 3 holds
( f is_hpartial_differentiable`23_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 Def7 defines is_hpartial_differentiable`31_in PDIFF_5:def_7_:_
for f being PartFunc of (REAL 3),REAL
for u being Element of REAL 3 holds
( f is_hpartial_differentiable`31_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 Def8 defines is_hpartial_differentiable`32_in PDIFF_5:def_8_:_
for f being PartFunc of (REAL 3),REAL
for u being Element of REAL 3 holds
( f is_hpartial_differentiable`32_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 Def9 defines is_hpartial_differentiable`33_in PDIFF_5:def_9_:_
for f being PartFunc of (REAL 3),REAL
for u being Element of REAL 3 holds
( f is_hpartial_differentiable`33_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_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)) ) ) ) );
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
consider x0, y0, z0 being Real such that
A2: ( 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)) ) ) by A1, Def1;
consider N being Neighbourhood of x0 such that
A3: ( 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)) ) by A2;
consider L being LinearFunc, R being RestFunc such that
A4: 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)) by A3;
consider r being Real such that
A5: for p being Real holds L . p = r * p by FDIFF_1:def_3;
take r ; ::_thesis: 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
( r = 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)) ) ) ) )
L . 1 = r * 1 by A5
.= r ;
hence 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
( r = 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)) ) ) ) ) by A2, A3, A4; ::_thesis: verum
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
let r, s be Real; ::_thesis: ( 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
( r = 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
( s = 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)) ) ) ) ) implies r = s )
assume that
A6: 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
( r = 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)) ) ) ) ) and
A7: 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
( s = 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)) ) ) ) ) ; ::_thesis: r = s
consider x0, y0, z0 being Real such that
A8: ( 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
( r = 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)) ) ) ) ) by A6;
consider N being Neighbourhood of x0 such that
A9: ( N c= dom (SVF1 (1,(pdiff1 (f,1)),u)) & ex L being LinearFunc ex R being RestFunc st
( r = 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)) ) ) ) by A8;
consider L being LinearFunc, R being RestFunc such that
A10: ( r = 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)) ) ) by A9;
consider x1, y1, z1 being Real such that
A11: ( u = <*x1,y1,z1*> & ex N being Neighbourhood of x1 st
( N c= dom (SVF1 (1,(pdiff1 (f,1)),u)) & ex L being LinearFunc ex R being RestFunc st
( s = L . 1 & ( for x being Real st x in N holds
((SVF1 (1,(pdiff1 (f,1)),u)) . x) - ((SVF1 (1,(pdiff1 (f,1)),u)) . x1) = (L . (x - x1)) + (R . (x - x1)) ) ) ) ) by A7;
consider N1 being Neighbourhood of x1 such that
A12: ( N1 c= dom (SVF1 (1,(pdiff1 (f,1)),u)) & ex L being LinearFunc ex R being RestFunc st
( s = L . 1 & ( for x being Real st x in N1 holds
((SVF1 (1,(pdiff1 (f,1)),u)) . x) - ((SVF1 (1,(pdiff1 (f,1)),u)) . x1) = (L . (x - x1)) + (R . (x - x1)) ) ) ) by A11;
consider L1 being LinearFunc, R1 being RestFunc such that
A13: ( s = L1 . 1 & ( for x being Real st x in N1 holds
((SVF1 (1,(pdiff1 (f,1)),u)) . x) - ((SVF1 (1,(pdiff1 (f,1)),u)) . x1) = (L1 . (x - x1)) + (R1 . (x - x1)) ) ) by A12;
consider r1 being Real such that
A14: for p being Real holds L . p = r1 * p by FDIFF_1:def_3;
consider p1 being Real such that
A15: for p being Real holds L1 . p = p1 * p by FDIFF_1:def_3;
A16: r = r1 * 1 by A10, A14;
A17: s = p1 * 1 by A13, A15;
A18: ( x0 = x1 & y0 = y1 & z0 = z1 ) by A8, A11, FINSEQ_1:78;
A19: now__::_thesis:_for_x_being_Real_st_x_in_N_&_x_in_N1_holds_
(r_*_(x_-_x0))_+_(R_._(x_-_x0))_=_(s_*_(x_-_x0))_+_(R1_._(x_-_x0))
let x be Real; ::_thesis: ( x in N & x in N1 implies (r * (x - x0)) + (R . (x - x0)) = (s * (x - x0)) + (R1 . (x - x0)) )
assume A20: ( x in N & x in N1 ) ; ::_thesis: (r * (x - x0)) + (R . (x - x0)) = (s * (x - x0)) + (R1 . (x - x0))
then ((SVF1 (1,(pdiff1 (f,1)),u)) . x) - ((SVF1 (1,(pdiff1 (f,1)),u)) . x0) = (L . (x - x0)) + (R . (x - x0)) by A10;
then (L . (x - x0)) + (R . (x - x0)) = (L1 . (x - x0)) + (R1 . (x - x0)) by A13, A18, A20;
then (r1 * (x - x0)) + (R . (x - x0)) = (L1 . (x - x0)) + (R1 . (x - x0)) by A14;
hence (r * (x - x0)) + (R . (x - x0)) = (s * (x - x0)) + (R1 . (x - x0)) by A15, A16, A17; ::_thesis: verum
end;
consider N0 being Neighbourhood of x0 such that
A21: ( N0 c= N & N0 c= N1 ) by A18, RCOMP_1:17;
consider g being real number such that
A22: ( 0 < g & N0 = ].(x0 - g),(x0 + g).[ ) by RCOMP_1:def_6;
deffunc H1( Element of NAT ) -> Element of REAL = g / ($1 + 2);
consider s1 being Real_Sequence such that
A23: for n being Element of NAT holds s1 . n = H1(n) from SEQ_1:sch_1();
now__::_thesis:_for_n_being_Element_of_NAT_holds_s1_._n_<>_0
let n be Element of NAT ; ::_thesis: s1 . n <> 0
g / (n + 2) <> 0 by A22, XREAL_1:139;
hence s1 . n <> 0 by A23; ::_thesis: verum
end;
then A24: s1 is non-zero by SEQ_1:5;
( s1 is convergent & lim s1 = 0 ) by A23, SEQ_4:31;
then reconsider h = s1 as non-zero 0 -convergent Real_Sequence by A24, FDIFF_1:def_1;
A25: for n being Element of NAT ex x being Real st
( x in N & x in N1 & h . n = x - x0 )
proof
let n be Element of NAT ; ::_thesis: ex x being Real st
( x in N & x in N1 & h . n = x - x0 )
A26: g / (n + 2) > 0 by A22, XREAL_1:139;
0 + 1 < (n + 1) + 1 by XREAL_1:6;
then g / (n + 2) < g / 1 by A22, XREAL_1:76;
then A27: x0 + (g / (n + 2)) < x0 + g by XREAL_1:6;
x0 + (- g) < x0 + (g / (n + 2)) by A22, A26, XREAL_1:6;
then A28: x0 + (g / (n + 2)) in ].(x0 - g),(x0 + g).[ by A27;
take x0 + (g / (n + 2)) ; ::_thesis: ( x0 + (g / (n + 2)) in N & x0 + (g / (n + 2)) in N1 & h . n = (x0 + (g / (n + 2))) - x0 )
thus ( x0 + (g / (n + 2)) in N & x0 + (g / (n + 2)) in N1 & h . n = (x0 + (g / (n + 2))) - x0 ) by A21, A22, A23, A28; ::_thesis: verum
end;
A29: for n being Nat holds r - s = (((h ") (#) (R1 /* h)) - ((h ") (#) (R /* h))) . n
proof
let n be Nat; ::_thesis: r - s = (((h ") (#) (R1 /* h)) - ((h ") (#) (R /* h))) . n
A30: n in NAT by ORDINAL1:def_12;
then ex x being Real st
( x in N & x in N1 & h . n = x - x0 ) by A25;
then A31: (r * (h . n)) + (R . (h . n)) = (s * (h . n)) + (R1 . (h . n)) by A19;
A32: h . n <> 0 by A30, SEQ_1:5;
A33: ((r * (h . n)) / (h . n)) + ((R . (h . n)) / (h . n)) = ((s * (h . n)) + (R1 . (h . n))) / (h . n) by A31, XCMPLX_1:62;
A34: (r * (h . n)) / (h . n) = r * ((h . n) / (h . n)) by XCMPLX_1:74
.= r * 1 by A32, XCMPLX_1:60
.= r ;
(s * (h . n)) / (h . n) = s * ((h . n) / (h . n)) by XCMPLX_1:74
.= s * 1 by A32, XCMPLX_1:60
.= s ;
then A35: r + ((R . (h . n)) / (h . n)) = s + ((R1 . (h . n)) / (h . n)) by A33, A34, XCMPLX_1:62;
dom R = REAL by PARTFUN1:def_2;
then A36: rng h c= dom R ;
dom R1 = REAL by PARTFUN1:def_2;
then A37: rng h c= dom R1 ;
A38: (R . (h . n)) / (h . n) = (R . (h . n)) * ((h . n) ") by XCMPLX_0:def_9
.= (R . (h . n)) * ((h ") . n) by VALUED_1:10
.= ((R /* h) . n) * ((h ") . n) by A36, A30, FUNCT_2:108
.= ((h ") (#) (R /* h)) . n by VALUED_1:5 ;
(R1 . (h . n)) / (h . n) = (R1 . (h . n)) * ((h . n) ") by XCMPLX_0:def_9
.= (R1 . (h . n)) * ((h ") . n) by VALUED_1:10
.= ((R1 /* h) . n) * ((h ") . n) by A37, A30, FUNCT_2:108
.= ((h ") (#) (R1 /* h)) . n by VALUED_1:5 ;
then r = s + ((((h ") (#) (R1 /* h)) . n) - (((h ") (#) (R /* h)) . n)) by A35, A38;
hence r - s = (((h ") (#) (R1 /* h)) - ((h ") (#) (R /* h))) . n by A30, RFUNCT_2:1; ::_thesis: verum
end;
then A39: ((h ") (#) (R1 /* h)) - ((h ") (#) (R /* h)) is constant by VALUED_0:def_18;
(((h ") (#) (R1 /* h)) - ((h ") (#) (R /* h))) . 1 = r - s by A29;
then A40: lim (((h ") (#) (R1 /* h)) - ((h ") (#) (R /* h))) = r - s by A39, SEQ_4:25;
A41: ( (h ") (#) (R /* h) is convergent & lim ((h ") (#) (R /* h)) = 0 ) by FDIFF_1:def_2;
( (h ") (#) (R1 /* h) is convergent & lim ((h ") (#) (R1 /* h)) = 0 ) by FDIFF_1:def_2;
then r - s = 0 - 0 by A40, A41, SEQ_2:12;
hence r = s ; ::_thesis: verum
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_differentiable`11_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_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)) ) ) ) );
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
consider x0, y0, z0 being Real such that
A2: ( 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)) ) ) by A1, Def2;
consider N being Neighbourhood of y0 such that
A3: ( 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)) ) by A2;
consider L being LinearFunc, R being RestFunc such that
A4: 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)) by A3;
consider r being Real such that
A5: for p being Real holds L . p = r * p by FDIFF_1:def_3;
take r ; ::_thesis: 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
( r = 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)) ) ) ) )
L . 1 = r * 1 by A5
.= r ;
hence 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
( r = 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)) ) ) ) ) by A2, A3, A4; ::_thesis: verum
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
let r, s be Real; ::_thesis: ( 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
( r = 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
( s = 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)) ) ) ) ) implies r = s )
assume that
A6: 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
( r = 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)) ) ) ) ) and
A7: 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
( s = 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)) ) ) ) ) ; ::_thesis: r = s
consider x0, y0, z0 being Real such that
A8: ( 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
( r = 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)) ) ) ) ) by A6;
consider N being Neighbourhood of y0 such that
A9: ( N c= dom (SVF1 (2,(pdiff1 (f,1)),u)) & ex L being LinearFunc ex R being RestFunc st
( r = 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)) ) ) ) by A8;
consider L being LinearFunc, R being RestFunc such that
A10: ( r = 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)) ) ) by A9;
consider x1, y1, z1 being Real such that
A11: ( u = <*x1,y1,z1*> & ex N being Neighbourhood of y1 st
( N c= dom (SVF1 (2,(pdiff1 (f,1)),u)) & ex L being LinearFunc ex R being RestFunc st
( s = L . 1 & ( for y being Real st y in N holds
((SVF1 (2,(pdiff1 (f,1)),u)) . y) - ((SVF1 (2,(pdiff1 (f,1)),u)) . y1) = (L . (y - y1)) + (R . (y - y1)) ) ) ) ) by A7;
consider N1 being Neighbourhood of y1 such that
A12: ( N1 c= dom (SVF1 (2,(pdiff1 (f,1)),u)) & ex L being LinearFunc ex R being RestFunc st
( s = L . 1 & ( for y being Real st y in N1 holds
((SVF1 (2,(pdiff1 (f,1)),u)) . y) - ((SVF1 (2,(pdiff1 (f,1)),u)) . y1) = (L . (y - y1)) + (R . (y - y1)) ) ) ) by A11;
consider L1 being LinearFunc, R1 being RestFunc such that
A13: ( s = L1 . 1 & ( for y being Real st y in N1 holds
((SVF1 (2,(pdiff1 (f,1)),u)) . y) - ((SVF1 (2,(pdiff1 (f,1)),u)) . y1) = (L1 . (y - y1)) + (R1 . (y - y1)) ) ) by A12;
consider r1 being Real such that
A14: for p being Real holds L . p = r1 * p by FDIFF_1:def_3;
consider p1 being Real such that
A15: for p being Real holds L1 . p = p1 * p by FDIFF_1:def_3;
A16: r = r1 * 1 by A10, A14;
A17: s = p1 * 1 by A13, A15;
A18: ( x0 = x1 & y0 = y1 & z0 = z1 ) by A8, A11, FINSEQ_1:78;
A19: now__::_thesis:_for_y_being_Real_st_y_in_N_&_y_in_N1_holds_
(r_*_(y_-_y0))_+_(R_._(y_-_y0))_=_(s_*_(y_-_y0))_+_(R1_._(y_-_y0))
let y be Real; ::_thesis: ( y in N & y in N1 implies (r * (y - y0)) + (R . (y - y0)) = (s * (y - y0)) + (R1 . (y - y0)) )
assume A20: ( y in N & y in N1 ) ; ::_thesis: (r * (y - y0)) + (R . (y - y0)) = (s * (y - y0)) + (R1 . (y - y0))
then ((SVF1 (2,(pdiff1 (f,1)),u)) . y) - ((SVF1 (2,(pdiff1 (f,1)),u)) . y0) = (L . (y - y0)) + (R . (y - y0)) by A10;
then (L . (y - y0)) + (R . (y - y0)) = (L1 . (y - y0)) + (R1 . (y - y0)) by A13, A18, A20;
then (r1 * (y - y0)) + (R . (y - y0)) = (L1 . (y - y0)) + (R1 . (y - y0)) by A14;
hence (r * (y - y0)) + (R . (y - y0)) = (s * (y - y0)) + (R1 . (y - y0)) by A15, A16, A17; ::_thesis: verum
end;
consider N0 being Neighbourhood of y0 such that
A21: ( N0 c= N & N0 c= N1 ) by A18, RCOMP_1:17;
consider g being real number such that
A22: ( 0 < g & N0 = ].(y0 - g),(y0 + g).[ ) by RCOMP_1:def_6;
deffunc H1( Element of NAT ) -> Element of REAL = g / ($1 + 2);
consider s1 being Real_Sequence such that
A23: for n being Element of NAT holds s1 . n = H1(n) from SEQ_1:sch_1();
now__::_thesis:_for_n_being_Element_of_NAT_holds_s1_._n_<>_0
let n be Element of NAT ; ::_thesis: s1 . n <> 0
g / (n + 2) <> 0 by A22, XREAL_1:139;
hence s1 . n <> 0 by A23; ::_thesis: verum
end;
then A24: s1 is non-zero by SEQ_1:5;
( s1 is convergent & lim s1 = 0 ) by A23, SEQ_4:31;
then reconsider h = s1 as non-zero 0 -convergent Real_Sequence by A24, FDIFF_1:def_1;
A25: for n being Element of NAT ex y being Real st
( y in N & y in N1 & h . n = y - y0 )
proof
let n be Element of NAT ; ::_thesis: ex y being Real st
( y in N & y in N1 & h . n = y - y0 )
A26: g / (n + 2) > 0 by A22, XREAL_1:139;
0 + 1 < (n + 1) + 1 by XREAL_1:6;
then g / (n + 2) < g / 1 by A22, XREAL_1:76;
then A27: y0 + (g / (n + 2)) < y0 + g by XREAL_1:6;
y0 + (- g) < y0 + (g / (n + 2)) by A22, A26, XREAL_1:6;
then A28: y0 + (g / (n + 2)) in ].(y0 - g),(y0 + g).[ by A27;
take y0 + (g / (n + 2)) ; ::_thesis: ( y0 + (g / (n + 2)) in N & y0 + (g / (n + 2)) in N1 & h . n = (y0 + (g / (n + 2))) - y0 )
thus ( y0 + (g / (n + 2)) in N & y0 + (g / (n + 2)) in N1 & h . n = (y0 + (g / (n + 2))) - y0 ) by A21, A22, A23, A28; ::_thesis: verum
end;
A29: now__::_thesis:_for_n_being_Nat_holds_r_-_s_=_(((h_")_(#)_(R1_/*_h))_-_((h_")_(#)_(R_/*_h)))_._n
let n be Nat; ::_thesis: r - s = (((h ") (#) (R1 /* h)) - ((h ") (#) (R /* h))) . n
A30: n in NAT by ORDINAL1:def_12;
then ex y being Real st
( y in N & y in N1 & h . n = y - y0 ) by A25;
then A31: (r * (h . n)) + (R . (h . n)) = (s * (h . n)) + (R1 . (h . n)) by A19;
A32: h . n <> 0 by A30, SEQ_1:5;
A33: ((r * (h . n)) / (h . n)) + ((R . (h . n)) / (h . n)) = ((s * (h . n)) + (R1 . (h . n))) / (h . n) by A31, XCMPLX_1:62;
A34: (r * (h . n)) / (h . n) = r * ((h . n) / (h . n)) by XCMPLX_1:74
.= r * 1 by A32, XCMPLX_1:60
.= r ;
(s * (h . n)) / (h . n) = s * ((h . n) / (h . n)) by XCMPLX_1:74
.= s * 1 by A32, XCMPLX_1:60
.= s ;
then A35: r + ((R . (h . n)) / (h . n)) = s + ((R1 . (h . n)) / (h . n)) by A33, A34, XCMPLX_1:62;
dom R = REAL by PARTFUN1:def_2;
then A36: rng h c= dom R ;
dom R1 = REAL by PARTFUN1:def_2;
then A37: rng h c= dom R1 ;
A38: (R . (h . n)) / (h . n) = (R . (h . n)) * ((h . n) ") by XCMPLX_0:def_9
.= (R . (h . n)) * ((h ") . n) by VALUED_1:10
.= ((R /* h) . n) * ((h ") . n) by A36, A30, FUNCT_2:108
.= ((h ") (#) (R /* h)) . n by VALUED_1:5 ;
(R1 . (h . n)) / (h . n) = (R1 . (h . n)) * ((h . n) ") by XCMPLX_0:def_9
.= (R1 . (h . n)) * ((h ") . n) by VALUED_1:10
.= ((R1 /* h) . n) * ((h ") . n) by A37, A30, FUNCT_2:108
.= ((h ") (#) (R1 /* h)) . n by VALUED_1:5 ;
then r = s + ((((h ") (#) (R1 /* h)) . n) - (((h ") (#) (R /* h)) . n)) by A35, A38;
hence r - s = (((h ") (#) (R1 /* h)) - ((h ") (#) (R /* h))) . n by A30, RFUNCT_2:1; ::_thesis: verum
end;
then A39: ((h ") (#) (R1 /* h)) - ((h ") (#) (R /* h)) is constant by VALUED_0:def_18;
(((h ") (#) (R1 /* h)) - ((h ") (#) (R /* h))) . 1 = r - s by A29;
then A40: lim (((h ") (#) (R1 /* h)) - ((h ") (#) (R /* h))) = r - s by A39, SEQ_4:25;
A41: ( (h ") (#) (R /* h) is convergent & lim ((h ") (#) (R /* h)) = 0 ) by FDIFF_1:def_2;
( (h ") (#) (R1 /* h) is convergent & lim ((h ") (#) (R1 /* h)) = 0 ) by FDIFF_1:def_2;
then r - s = 0 - 0 by A40, A41, SEQ_2:12;
hence r = s ; ::_thesis: verum
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_differentiable`12_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_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)) ) ) ) );
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
consider x0, y0, z0 being Real such that
A2: ( 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)) ) ) by A1, Def3;
consider N being Neighbourhood of z0 such that
A3: ( 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)) ) by A2;
consider L being LinearFunc, R being RestFunc such that
A4: 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)) by A3;
consider r being Real such that
A5: for p being Real holds L . p = r * p by FDIFF_1:def_3;
take r ; ::_thesis: 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
( r = 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)) ) ) ) )
L . 1 = r * 1 by A5
.= r ;
hence 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
( r = 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)) ) ) ) ) by A2, A3, A4; ::_thesis: verum
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
let r, s be Real; ::_thesis: ( 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
( r = 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
( s = 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)) ) ) ) ) implies r = s )
assume that
A6: 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
( r = 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)) ) ) ) ) and
A7: 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
( s = 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)) ) ) ) ) ; ::_thesis: r = s
consider x0, y0, z0 being Real such that
A8: ( 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
( r = 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)) ) ) ) ) by A6;
consider N being Neighbourhood of z0 such that
A9: ( N c= dom (SVF1 (3,(pdiff1 (f,1)),u)) & ex L being LinearFunc ex R being RestFunc st
( r = 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)) ) ) ) by A8;
consider L being LinearFunc, R being RestFunc such that
A10: ( r = 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)) ) ) by A9;
consider x1, y1, z1 being Real such that
A11: ( u = <*x1,y1,z1*> & ex N being Neighbourhood of z1 st
( N c= dom (SVF1 (3,(pdiff1 (f,1)),u)) & ex L being LinearFunc ex R being RestFunc st
( s = L . 1 & ( for z being Real st z in N holds
((SVF1 (3,(pdiff1 (f,1)),u)) . z) - ((SVF1 (3,(pdiff1 (f,1)),u)) . z1) = (L . (z - z1)) + (R . (z - z1)) ) ) ) ) by A7;
consider N1 being Neighbourhood of z1 such that
A12: ( N1 c= dom (SVF1 (3,(pdiff1 (f,1)),u)) & ex L being LinearFunc ex R being RestFunc st
( s = L . 1 & ( for z being Real st z in N1 holds
((SVF1 (3,(pdiff1 (f,1)),u)) . z) - ((SVF1 (3,(pdiff1 (f,1)),u)) . z1) = (L . (z - z1)) + (R . (z - z1)) ) ) ) by A11;
consider L1 being LinearFunc, R1 being RestFunc such that
A13: ( s = L1 . 1 & ( for z being Real st z in N1 holds
((SVF1 (3,(pdiff1 (f,1)),u)) . z) - ((SVF1 (3,(pdiff1 (f,1)),u)) . z1) = (L1 . (z - z1)) + (R1 . (z - z1)) ) ) by A12;
consider r1 being Real such that
A14: for p being Real holds L . p = r1 * p by FDIFF_1:def_3;
consider p1 being Real such that
A15: for p being Real holds L1 . p = p1 * p by FDIFF_1:def_3;
A16: r = r1 * 1 by A10, A14;
A17: s = p1 * 1 by A13, A15;
A18: ( x0 = x1 & y0 = y1 & z0 = z1 ) by A8, A11, FINSEQ_1:78;
A19: now__::_thesis:_for_z_being_Real_st_z_in_N_&_z_in_N1_holds_
(r_*_(z_-_z0))_+_(R_._(z_-_z0))_=_(s_*_(z_-_z0))_+_(R1_._(z_-_z0))
let z be Real; ::_thesis: ( z in N & z in N1 implies (r * (z - z0)) + (R . (z - z0)) = (s * (z - z0)) + (R1 . (z - z0)) )
assume A20: ( z in N & z in N1 ) ; ::_thesis: (r * (z - z0)) + (R . (z - z0)) = (s * (z - z0)) + (R1 . (z - z0))
then ((SVF1 (3,(pdiff1 (f,1)),u)) . z) - ((SVF1 (3,(pdiff1 (f,1)),u)) . z0) = (L . (z - z0)) + (R . (z - z0)) by A10;
then (L . (z - z0)) + (R . (z - z0)) = (L1 . (z - z0)) + (R1 . (z - z0)) by A13, A18, A20;
then (r1 * (z - z0)) + (R . (z - z0)) = (L1 . (z - z0)) + (R1 . (z - z0)) by A14;
hence (r * (z - z0)) + (R . (z - z0)) = (s * (z - z0)) + (R1 . (z - z0)) by A15, A16, A17; ::_thesis: verum
end;
consider N0 being Neighbourhood of z0 such that
A21: ( N0 c= N & N0 c= N1 ) by A18, RCOMP_1:17;
consider g being real number such that
A22: ( 0 < g & N0 = ].(z0 - g),(z0 + g).[ ) by RCOMP_1:def_6;
deffunc H1( Element of NAT ) -> Element of REAL = g / ($1 + 2);
consider s1 being Real_Sequence such that
A23: for n being Element of NAT holds s1 . n = H1(n) from SEQ_1:sch_1();
now__::_thesis:_for_n_being_Element_of_NAT_holds_s1_._n_<>_0
let n be Element of NAT ; ::_thesis: s1 . n <> 0
g / (n + 2) <> 0 by A22, XREAL_1:139;
hence s1 . n <> 0 by A23; ::_thesis: verum
end;
then A24: s1 is non-zero by SEQ_1:5;
( s1 is convergent & lim s1 = 0 ) by A23, SEQ_4:31;
then reconsider h = s1 as non-zero 0 -convergent Real_Sequence by A24, FDIFF_1:def_1;
A25: for n being Element of NAT ex z being Real st
( z in N & z in N1 & h . n = z - z0 )
proof
let n be Element of NAT ; ::_thesis: ex z being Real st
( z in N & z in N1 & h . n = z - z0 )
A26: g / (n + 2) > 0 by A22, XREAL_1:139;
0 + 1 < (n + 1) + 1 by XREAL_1:6;
then g / (n + 2) < g / 1 by A22, XREAL_1:76;
then A27: z0 + (g / (n + 2)) < z0 + g by XREAL_1:6;
z0 + (- g) < z0 + (g / (n + 2)) by A22, A26, XREAL_1:6;
then A28: z0 + (g / (n + 2)) in ].(z0 - g),(z0 + g).[ by A27;
take z0 + (g / (n + 2)) ; ::_thesis: ( z0 + (g / (n + 2)) in N & z0 + (g / (n + 2)) in N1 & h . n = (z0 + (g / (n + 2))) - z0 )
thus ( z0 + (g / (n + 2)) in N & z0 + (g / (n + 2)) in N1 & h . n = (z0 + (g / (n + 2))) - z0 ) by A21, A22, A23, A28; ::_thesis: verum
end;
A29: now__::_thesis:_for_n_being_Nat_holds_r_-_s_=_(((h_")_(#)_(R1_/*_h))_-_((h_")_(#)_(R_/*_h)))_._n
let n be Nat; ::_thesis: r - s = (((h ") (#) (R1 /* h)) - ((h ") (#) (R /* h))) . n
A30: n in NAT by ORDINAL1:def_12;
then ex z being Real st
( z in N & z in N1 & h . n = z - z0 ) by A25;
then A31: (r * (h . n)) + (R . (h . n)) = (s * (h . n)) + (R1 . (h . n)) by A19;
A32: h . n <> 0 by A30, SEQ_1:5;
A33: ((r * (h . n)) / (h . n)) + ((R . (h . n)) / (h . n)) = ((s * (h . n)) + (R1 . (h . n))) / (h . n) by A31, XCMPLX_1:62;
A34: (r * (h . n)) / (h . n) = r * ((h . n) / (h . n)) by XCMPLX_1:74
.= r * 1 by A32, XCMPLX_1:60
.= r ;
(s * (h . n)) / (h . n) = s * ((h . n) / (h . n)) by XCMPLX_1:74
.= s * 1 by A32, XCMPLX_1:60
.= s ;
then A35: r + ((R . (h . n)) / (h . n)) = s + ((R1 . (h . n)) / (h . n)) by A33, A34, XCMPLX_1:62;
dom R = REAL by PARTFUN1:def_2;
then A36: rng h c= dom R ;
dom R1 = REAL by PARTFUN1:def_2;
then A37: rng h c= dom R1 ;
A38: (R . (h . n)) / (h . n) = (R . (h . n)) * ((h . n) ") by XCMPLX_0:def_9
.= (R . (h . n)) * ((h ") . n) by VALUED_1:10
.= ((R /* h) . n) * ((h ") . n) by A36, A30, FUNCT_2:108
.= ((h ") (#) (R /* h)) . n by VALUED_1:5 ;
(R1 . (h . n)) / (h . n) = (R1 . (h . n)) * ((h . n) ") by XCMPLX_0:def_9
.= (R1 . (h . n)) * ((h ") . n) by VALUED_1:10
.= ((R1 /* h) . n) * ((h ") . n) by A37, A30, FUNCT_2:108
.= ((h ") (#) (R1 /* h)) . n by VALUED_1:5 ;
then r = s + ((((h ") (#) (R1 /* h)) . n) - (((h ") (#) (R /* h)) . n)) by A35, A38;
hence r - s = (((h ") (#) (R1 /* h)) - ((h ") (#) (R /* h))) . n by A30, RFUNCT_2:1; ::_thesis: verum
end;
then A39: ((h ") (#) (R1 /* h)) - ((h ") (#) (R /* h)) is constant by VALUED_0:def_18;
(((h ") (#) (R1 /* h)) - ((h ") (#) (R /* h))) . 1 = r - s by A29;
then A40: lim (((h ") (#) (R1 /* h)) - ((h ") (#) (R /* h))) = r - s by A39, SEQ_4:25;
A41: ( (h ") (#) (R /* h) is convergent & lim ((h ") (#) (R /* h)) = 0 ) by FDIFF_1:def_2;
( (h ") (#) (R1 /* h) is convergent & lim ((h ") (#) (R1 /* h)) = 0 ) by FDIFF_1:def_2;
then r - s = 0 - 0 by A40, A41, SEQ_2:12;
hence r = s ; ::_thesis: verum
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_differentiable`13_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_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)) ) ) ) );
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
consider x0, y0, z0 being Real such that
A2: ( 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)) ) ) by A1, Def4;
consider N being Neighbourhood of x0 such that
A3: ( 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)) ) by A2;
consider L being LinearFunc, R being RestFunc such that
A4: 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)) by A3;
consider r being Real such that
A5: for p being Real holds L . p = r * p by FDIFF_1:def_3;
take r ; ::_thesis: 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
( r = 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)) ) ) ) )
L . 1 = r * 1 by A5
.= r ;
hence 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
( r = 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)) ) ) ) ) by A2, A3, A4; ::_thesis: verum
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
let r, s be Real; ::_thesis: ( 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
( r = 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
( s = 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)) ) ) ) ) implies r = s )
assume that
A6: 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
( r = 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)) ) ) ) ) and
A7: 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
( s = 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)) ) ) ) ) ; ::_thesis: r = s
consider x0, y0, z0 being Real such that
A8: ( 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
( r = 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)) ) ) ) ) by A6;
consider N being Neighbourhood of x0 such that
A9: ( N c= dom (SVF1 (1,(pdiff1 (f,2)),u)) & ex L being LinearFunc ex R being RestFunc st
( r = 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)) ) ) ) by A8;
consider L being LinearFunc, R being RestFunc such that
A10: ( r = 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)) ) ) by A9;
consider x1, y1, z1 being Real such that
A11: ( u = <*x1,y1,z1*> & ex N being Neighbourhood of x1 st
( N c= dom (SVF1 (1,(pdiff1 (f,2)),u)) & ex L being LinearFunc ex R being RestFunc st
( s = L . 1 & ( for x being Real st x in N holds
((SVF1 (1,(pdiff1 (f,2)),u)) . x) - ((SVF1 (1,(pdiff1 (f,2)),u)) . x1) = (L . (x - x1)) + (R . (x - x1)) ) ) ) ) by A7;
consider N1 being Neighbourhood of x1 such that
A12: ( N1 c= dom (SVF1 (1,(pdiff1 (f,2)),u)) & ex L being LinearFunc ex R being RestFunc st
( s = L . 1 & ( for x being Real st x in N1 holds
((SVF1 (1,(pdiff1 (f,2)),u)) . x) - ((SVF1 (1,(pdiff1 (f,2)),u)) . x1) = (L . (x - x1)) + (R . (x - x1)) ) ) ) by A11;
consider L1 being LinearFunc, R1 being RestFunc such that
A13: ( s = L1 . 1 & ( for x being Real st x in N1 holds
((SVF1 (1,(pdiff1 (f,2)),u)) . x) - ((SVF1 (1,(pdiff1 (f,2)),u)) . x1) = (L1 . (x - x1)) + (R1 . (x - x1)) ) ) by A12;
consider r1 being Real such that
A14: for p being Real holds L . p = r1 * p by FDIFF_1:def_3;
consider p1 being Real such that
A15: for p being Real holds L1 . p = p1 * p by FDIFF_1:def_3;
A16: r = r1 * 1 by A10, A14;
A17: s = p1 * 1 by A13, A15;
A18: ( x0 = x1 & y0 = y1 & z0 = z1 ) by A8, A11, FINSEQ_1:78;
A19: now__::_thesis:_for_x_being_Real_st_x_in_N_&_x_in_N1_holds_
(r_*_(x_-_x0))_+_(R_._(x_-_x0))_=_(s_*_(x_-_x0))_+_(R1_._(x_-_x0))
let x be Real; ::_thesis: ( x in N & x in N1 implies (r * (x - x0)) + (R . (x - x0)) = (s * (x - x0)) + (R1 . (x - x0)) )
assume A20: ( x in N & x in N1 ) ; ::_thesis: (r * (x - x0)) + (R . (x - x0)) = (s * (x - x0)) + (R1 . (x - x0))
then ((SVF1 (1,(pdiff1 (f,2)),u)) . x) - ((SVF1 (1,(pdiff1 (f,2)),u)) . x0) = (L . (x - x0)) + (R . (x - x0)) by A10;
then (L . (x - x0)) + (R . (x - x0)) = (L1 . (x - x0)) + (R1 . (x - x0)) by A13, A18, A20;
then (r1 * (x - x0)) + (R . (x - x0)) = (L1 . (x - x0)) + (R1 . (x - x0)) by A14;
hence (r * (x - x0)) + (R . (x - x0)) = (s * (x - x0)) + (R1 . (x - x0)) by A15, A16, A17; ::_thesis: verum
end;
consider N0 being Neighbourhood of x0 such that
A21: ( N0 c= N & N0 c= N1 ) by A18, RCOMP_1:17;
consider g being real number such that
A22: ( 0 < g & N0 = ].(x0 - g),(x0 + g).[ ) by RCOMP_1:def_6;
deffunc H1( Element of NAT ) -> Element of REAL = g / ($1 + 2);
consider s1 being Real_Sequence such that
A23: for n being Element of NAT holds s1 . n = H1(n) from SEQ_1:sch_1();
now__::_thesis:_for_n_being_Element_of_NAT_holds_s1_._n_<>_0
let n be Element of NAT ; ::_thesis: s1 . n <> 0
g / (n + 2) <> 0 by A22, XREAL_1:139;
hence s1 . n <> 0 by A23; ::_thesis: verum
end;
then A24: s1 is non-zero by SEQ_1:5;
( s1 is convergent & lim s1 = 0 ) by A23, SEQ_4:31;
then reconsider h = s1 as non-zero 0 -convergent Real_Sequence by A24, FDIFF_1:def_1;
A25: for n being Element of NAT ex x being Real st
( x in N & x in N1 & h . n = x - x0 )
proof
let n be Element of NAT ; ::_thesis: ex x being Real st
( x in N & x in N1 & h . n = x - x0 )
A26: g / (n + 2) > 0 by A22, XREAL_1:139;
0 + 1 < (n + 1) + 1 by XREAL_1:6;
then g / (n + 2) < g / 1 by A22, XREAL_1:76;
then A27: x0 + (g / (n + 2)) < x0 + g by XREAL_1:6;
x0 + (- g) < x0 + (g / (n + 2)) by A22, A26, XREAL_1:6;
then A28: x0 + (g / (n + 2)) in ].(x0 - g),(x0 + g).[ by A27;
take x0 + (g / (n + 2)) ; ::_thesis: ( x0 + (g / (n + 2)) in N & x0 + (g / (n + 2)) in N1 & h . n = (x0 + (g / (n + 2))) - x0 )
thus ( x0 + (g / (n + 2)) in N & x0 + (g / (n + 2)) in N1 & h . n = (x0 + (g / (n + 2))) - x0 ) by A21, A22, A23, A28; ::_thesis: verum
end;
A29: now__::_thesis:_for_n_being_Nat_holds_r_-_s_=_(((h_")_(#)_(R1_/*_h))_-_((h_")_(#)_(R_/*_h)))_._n
let n be Nat; ::_thesis: r - s = (((h ") (#) (R1 /* h)) - ((h ") (#) (R /* h))) . n
A30: n in NAT by ORDINAL1:def_12;
then ex x being Real st
( x in N & x in N1 & h . n = x - x0 ) by A25;
then A31: (r * (h . n)) + (R . (h . n)) = (s * (h . n)) + (R1 . (h . n)) by A19;
A32: h . n <> 0 by A30, SEQ_1:5;
A33: ((r * (h . n)) / (h . n)) + ((R . (h . n)) / (h . n)) = ((s * (h . n)) + (R1 . (h . n))) / (h . n) by A31, XCMPLX_1:62;
A34: (r * (h . n)) / (h . n) = r * ((h . n) / (h . n)) by XCMPLX_1:74
.= r * 1 by A32, XCMPLX_1:60
.= r ;
(s * (h . n)) / (h . n) = s * ((h . n) / (h . n)) by XCMPLX_1:74
.= s * 1 by A32, XCMPLX_1:60
.= s ;
then A35: r + ((R . (h . n)) / (h . n)) = s + ((R1 . (h . n)) / (h . n)) by A33, A34, XCMPLX_1:62;
dom R = REAL by PARTFUN1:def_2;
then A36: rng h c= dom R ;
dom R1 = REAL by PARTFUN1:def_2;
then A37: rng h c= dom R1 ;
A38: (R . (h . n)) / (h . n) = (R . (h . n)) * ((h . n) ") by XCMPLX_0:def_9
.= (R . (h . n)) * ((h ") . n) by VALUED_1:10
.= ((R /* h) . n) * ((h ") . n) by A36, A30, FUNCT_2:108
.= ((h ") (#) (R /* h)) . n by VALUED_1:5 ;
(R1 . (h . n)) / (h . n) = (R1 . (h . n)) * ((h . n) ") by XCMPLX_0:def_9
.= (R1 . (h . n)) * ((h ") . n) by VALUED_1:10
.= ((R1 /* h) . n) * ((h ") . n) by A37, A30, FUNCT_2:108
.= ((h ") (#) (R1 /* h)) . n by VALUED_1:5 ;
then r = s + ((((h ") (#) (R1 /* h)) . n) - (((h ") (#) (R /* h)) . n)) by A35, A38;
hence r - s = (((h ") (#) (R1 /* h)) - ((h ") (#) (R /* h))) . n by A30, VALUED_1:15; ::_thesis: verum
end;
then A39: ((h ") (#) (R1 /* h)) - ((h ") (#) (R /* h)) is constant by VALUED_0:def_18;
(((h ") (#) (R1 /* h)) - ((h ") (#) (R /* h))) . 1 = r - s by A29;
then A40: lim (((h ") (#) (R1 /* h)) - ((h ") (#) (R /* h))) = r - s by A39, SEQ_4:25;
A41: ( (h ") (#) (R /* h) is convergent & lim ((h ") (#) (R /* h)) = 0 ) by FDIFF_1:def_2;
( (h ") (#) (R1 /* h) is convergent & lim ((h ") (#) (R1 /* h)) = 0 ) by FDIFF_1:def_2;
then r - s = 0 - 0 by A40, A41, SEQ_2:12;
hence r = s ; ::_thesis: verum
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_differentiable`21_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_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)) ) ) ) );
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
consider x0, y0, z0 being Real such that
A2: ( 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)) ) ) by A1, Def5;
consider N being Neighbourhood of y0 such that
A3: ( 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)) ) by A2;
consider L being LinearFunc, R being RestFunc such that
A4: 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)) by A3;
consider r being Real such that
A5: for p being Real holds L . p = r * p by FDIFF_1:def_3;
take r ; ::_thesis: 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
( r = 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)) ) ) ) )
L . 1 = r * 1 by A5
.= r ;
hence 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
( r = 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)) ) ) ) ) by A2, A3, A4; ::_thesis: verum
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
let r, s be Real; ::_thesis: ( 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
( r = 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
( s = 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)) ) ) ) ) implies r = s )
assume that
A6: 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
( r = 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)) ) ) ) ) and
A7: 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
( s = 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)) ) ) ) ) ; ::_thesis: r = s
consider x0, y0, z0 being Real such that
A8: ( 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
( r = 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)) ) ) ) ) by A6;
consider N being Neighbourhood of y0 such that
A9: ( N c= dom (SVF1 (2,(pdiff1 (f,2)),u)) & ex L being LinearFunc ex R being RestFunc st
( r = 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)) ) ) ) by A8;
consider L being LinearFunc, R being RestFunc such that
A10: ( r = 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)) ) ) by A9;
consider x1, y1, z1 being Real such that
A11: ( u = <*x1,y1,z1*> & ex N being Neighbourhood of y1 st
( N c= dom (SVF1 (2,(pdiff1 (f,2)),u)) & ex L being LinearFunc ex R being RestFunc st
( s = L . 1 & ( for y being Real st y in N holds
((SVF1 (2,(pdiff1 (f,2)),u)) . y) - ((SVF1 (2,(pdiff1 (f,2)),u)) . y1) = (L . (y - y1)) + (R . (y - y1)) ) ) ) ) by A7;
consider N1 being Neighbourhood of y1 such that
A12: ( N1 c= dom (SVF1 (2,(pdiff1 (f,2)),u)) & ex L being LinearFunc ex R being RestFunc st
( s = L . 1 & ( for y being Real st y in N1 holds
((SVF1 (2,(pdiff1 (f,2)),u)) . y) - ((SVF1 (2,(pdiff1 (f,2)),u)) . y1) = (L . (y - y1)) + (R . (y - y1)) ) ) ) by A11;
consider L1 being LinearFunc, R1 being RestFunc such that
A13: ( s = L1 . 1 & ( for y being Real st y in N1 holds
((SVF1 (2,(pdiff1 (f,2)),u)) . y) - ((SVF1 (2,(pdiff1 (f,2)),u)) . y1) = (L1 . (y - y1)) + (R1 . (y - y1)) ) ) by A12;
consider r1 being Real such that
A14: for p being Real holds L . p = r1 * p by FDIFF_1:def_3;
consider p1 being Real such that
A15: for p being Real holds L1 . p = p1 * p by FDIFF_1:def_3;
A16: r = r1 * 1 by A10, A14;
A17: s = p1 * 1 by A13, A15;
A18: ( x0 = x1 & y0 = y1 & z0 = z1 ) by A8, A11, FINSEQ_1:78;
A19: now__::_thesis:_for_y_being_Real_st_y_in_N_&_y_in_N1_holds_
(r_*_(y_-_y0))_+_(R_._(y_-_y0))_=_(s_*_(y_-_y0))_+_(R1_._(y_-_y0))
let y be Real; ::_thesis: ( y in N & y in N1 implies (r * (y - y0)) + (R . (y - y0)) = (s * (y - y0)) + (R1 . (y - y0)) )
assume A20: ( y in N & y in N1 ) ; ::_thesis: (r * (y - y0)) + (R . (y - y0)) = (s * (y - y0)) + (R1 . (y - y0))
then ((SVF1 (2,(pdiff1 (f,2)),u)) . y) - ((SVF1 (2,(pdiff1 (f,2)),u)) . y0) = (L . (y - y0)) + (R . (y - y0)) by A10;
then (L . (y - y0)) + (R . (y - y0)) = (L1 . (y - y0)) + (R1 . (y - y0)) by A13, A18, A20;
then (r1 * (y - y0)) + (R . (y - y0)) = (L1 . (y - y0)) + (R1 . (y - y0)) by A14;
hence (r * (y - y0)) + (R . (y - y0)) = (s * (y - y0)) + (R1 . (y - y0)) by A15, A16, A17; ::_thesis: verum
end;
consider N0 being Neighbourhood of y0 such that
A21: ( N0 c= N & N0 c= N1 ) by A18, RCOMP_1:17;
consider g being real number such that
A22: ( 0 < g & N0 = ].(y0 - g),(y0 + g).[ ) by RCOMP_1:def_6;
deffunc H1( Element of NAT ) -> Element of REAL = g / ($1 + 2);
consider s1 being Real_Sequence such that
A23: for n being Element of NAT holds s1 . n = H1(n) from SEQ_1:sch_1();
now__::_thesis:_for_n_being_Element_of_NAT_holds_s1_._n_<>_0
let n be Element of NAT ; ::_thesis: s1 . n <> 0
g / (n + 2) <> 0 by A22, XREAL_1:139;
hence s1 . n <> 0 by A23; ::_thesis: verum
end;
then A24: s1 is non-zero by SEQ_1:5;
( s1 is convergent & lim s1 = 0 ) by A23, SEQ_4:31;
then reconsider h = s1 as non-zero 0 -convergent Real_Sequence by A24, FDIFF_1:def_1;
A25: for n being Element of NAT ex y being Real st
( y in N & y in N1 & h . n = y - y0 )
proof
let n be Element of NAT ; ::_thesis: ex y being Real st
( y in N & y in N1 & h . n = y - y0 )
A26: g / (n + 2) > 0 by A22, XREAL_1:139;
0 + 1 < (n + 1) + 1 by XREAL_1:6;
then g / (n + 2) < g / 1 by A22, XREAL_1:76;
then A27: y0 + (g / (n + 2)) < y0 + g by XREAL_1:6;
y0 + (- g) < y0 + (g / (n + 2)) by A22, A26, XREAL_1:6;
then A28: y0 + (g / (n + 2)) in ].(y0 - g),(y0 + g).[ by A27;
take y0 + (g / (n + 2)) ; ::_thesis: ( y0 + (g / (n + 2)) in N & y0 + (g / (n + 2)) in N1 & h . n = (y0 + (g / (n + 2))) - y0 )
thus ( y0 + (g / (n + 2)) in N & y0 + (g / (n + 2)) in N1 & h . n = (y0 + (g / (n + 2))) - y0 ) by A21, A22, A23, A28; ::_thesis: verum
end;
A29: now__::_thesis:_for_n_being_Nat_holds_r_-_s_=_(((h_")_(#)_(R1_/*_h))_-_((h_")_(#)_(R_/*_h)))_._n
let n be Nat; ::_thesis: r - s = (((h ") (#) (R1 /* h)) - ((h ") (#) (R /* h))) . n
A30: n in NAT by ORDINAL1:def_12;
then ex y being Real st
( y in N & y in N1 & h . n = y - y0 ) by A25;
then A31: (r * (h . n)) + (R . (h . n)) = (s * (h . n)) + (R1 . (h . n)) by A19;
A32: h . n <> 0 by A30, SEQ_1:5;
A33: ((r * (h . n)) / (h . n)) + ((R . (h . n)) / (h . n)) = ((s * (h . n)) + (R1 . (h . n))) / (h . n) by A31, XCMPLX_1:62;
A34: (r * (h . n)) / (h . n) = r * ((h . n) / (h . n)) by XCMPLX_1:74
.= r * 1 by A32, XCMPLX_1:60
.= r ;
(s * (h . n)) / (h . n) = s * ((h . n) / (h . n)) by XCMPLX_1:74
.= s * 1 by A32, XCMPLX_1:60
.= s ;
then A35: r + ((R . (h . n)) / (h . n)) = s + ((R1 . (h . n)) / (h . n)) by A33, A34, XCMPLX_1:62;
dom R = REAL by PARTFUN1:def_2;
then A36: rng h c= dom R ;
dom R1 = REAL by PARTFUN1:def_2;
then A37: rng h c= dom R1 ;
A38: (R . (h . n)) / (h . n) = (R . (h . n)) * ((h . n) ") by XCMPLX_0:def_9
.= (R . (h . n)) * ((h ") . n) by VALUED_1:10
.= ((R /* h) . n) * ((h ") . n) by A36, A30, FUNCT_2:108
.= ((h ") (#) (R /* h)) . n by VALUED_1:5 ;
(R1 . (h . n)) / (h . n) = (R1 . (h . n)) * ((h . n) ") by XCMPLX_0:def_9
.= (R1 . (h . n)) * ((h ") . n) by VALUED_1:10
.= ((R1 /* h) . n) * ((h ") . n) by A37, A30, FUNCT_2:108
.= ((h ") (#) (R1 /* h)) . n by VALUED_1:5 ;
then r = s + ((((h ") (#) (R1 /* h)) . n) - (((h ") (#) (R /* h)) . n)) by A35, A38;
hence r - s = (((h ") (#) (R1 /* h)) - ((h ") (#) (R /* h))) . n by A30, RFUNCT_2:1; ::_thesis: verum
end;
then A39: ((h ") (#) (R1 /* h)) - ((h ") (#) (R /* h)) is constant by VALUED_0:def_18;
(((h ") (#) (R1 /* h)) - ((h ") (#) (R /* h))) . 1 = r - s by A29;
then A40: lim (((h ") (#) (R1 /* h)) - ((h ") (#) (R /* h))) = r - s by A39, SEQ_4:25;
A41: ( (h ") (#) (R /* h) is convergent & lim ((h ") (#) (R /* h)) = 0 ) by FDIFF_1:def_2;
( (h ") (#) (R1 /* h) is convergent & lim ((h ") (#) (R1 /* h)) = 0 ) by FDIFF_1:def_2;
then r - s = 0 - 0 by A40, A41, SEQ_2:12;
hence r = s ; ::_thesis: verum
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_differentiable`22_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_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)) ) ) ) );
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
consider x0, y0, z0 being Real such that
A2: ( 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)) ) ) by A1, Def6;
consider N being Neighbourhood of z0 such that
A3: ( 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)) ) by A2;
consider L being LinearFunc, R being RestFunc such that
A4: 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)) by A3;
consider r being Real such that
A5: for p being Real holds L . p = r * p by FDIFF_1:def_3;
take r ; ::_thesis: 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
( r = 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)) ) ) ) )
L . 1 = r * 1 by A5
.= r ;
hence 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
( r = 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)) ) ) ) ) by A2, A3, A4; ::_thesis: verum
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
let r, s be Real; ::_thesis: ( 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
( r = 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
( s = 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)) ) ) ) ) implies r = s )
assume that
A6: 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
( r = 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)) ) ) ) ) and
A7: 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
( s = 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)) ) ) ) ) ; ::_thesis: r = s
consider x0, y0, z0 being Real such that
A8: ( 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
( r = 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)) ) ) ) ) by A6;
consider N being Neighbourhood of z0 such that
A9: ( N c= dom (SVF1 (3,(pdiff1 (f,2)),u)) & ex L being LinearFunc ex R being RestFunc st
( r = 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)) ) ) ) by A8;
consider L being LinearFunc, R being RestFunc such that
A10: ( r = 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)) ) ) by A9;
consider x1, y1, z1 being Real such that
A11: ( u = <*x1,y1,z1*> & ex N being Neighbourhood of z1 st
( N c= dom (SVF1 (3,(pdiff1 (f,2)),u)) & ex L being LinearFunc ex R being RestFunc st
( s = L . 1 & ( for z being Real st z in N holds
((SVF1 (3,(pdiff1 (f,2)),u)) . z) - ((SVF1 (3,(pdiff1 (f,2)),u)) . z1) = (L . (z - z1)) + (R . (z - z1)) ) ) ) ) by A7;
consider N1 being Neighbourhood of z1 such that
A12: ( N1 c= dom (SVF1 (3,(pdiff1 (f,2)),u)) & ex L being LinearFunc ex R being RestFunc st
( s = L . 1 & ( for z being Real st z in N1 holds
((SVF1 (3,(pdiff1 (f,2)),u)) . z) - ((SVF1 (3,(pdiff1 (f,2)),u)) . z1) = (L . (z - z1)) + (R . (z - z1)) ) ) ) by A11;
consider L1 being LinearFunc, R1 being RestFunc such that
A13: ( s = L1 . 1 & ( for z being Real st z in N1 holds
((SVF1 (3,(pdiff1 (f,2)),u)) . z) - ((SVF1 (3,(pdiff1 (f,2)),u)) . z1) = (L1 . (z - z1)) + (R1 . (z - z1)) ) ) by A12;
consider r1 being Real such that
A14: for p being Real holds L . p = r1 * p by FDIFF_1:def_3;
consider p1 being Real such that
A15: for p being Real holds L1 . p = p1 * p by FDIFF_1:def_3;
A16: r = r1 * 1 by A10, A14;
A17: s = p1 * 1 by A13, A15;
A18: ( x0 = x1 & y0 = y1 & z0 = z1 ) by A8, A11, FINSEQ_1:78;
A19: now__::_thesis:_for_z_being_Real_st_z_in_N_&_z_in_N1_holds_
(r_*_(z_-_z0))_+_(R_._(z_-_z0))_=_(s_*_(z_-_z0))_+_(R1_._(z_-_z0))
let z be Real; ::_thesis: ( z in N & z in N1 implies (r * (z - z0)) + (R . (z - z0)) = (s * (z - z0)) + (R1 . (z - z0)) )
assume A20: ( z in N & z in N1 ) ; ::_thesis: (r * (z - z0)) + (R . (z - z0)) = (s * (z - z0)) + (R1 . (z - z0))
then ((SVF1 (3,(pdiff1 (f,2)),u)) . z) - ((SVF1 (3,(pdiff1 (f,2)),u)) . z0) = (L . (z - z0)) + (R . (z - z0)) by A10;
then (L . (z - z0)) + (R . (z - z0)) = (L1 . (z - z0)) + (R1 . (z - z0)) by A13, A18, A20;
then (r1 * (z - z0)) + (R . (z - z0)) = (L1 . (z - z0)) + (R1 . (z - z0)) by A14;
hence (r * (z - z0)) + (R . (z - z0)) = (s * (z - z0)) + (R1 . (z - z0)) by A15, A16, A17; ::_thesis: verum
end;
consider N0 being Neighbourhood of z0 such that
A21: ( N0 c= N & N0 c= N1 ) by A18, RCOMP_1:17;
consider g being real number such that
A22: ( 0 < g & N0 = ].(z0 - g),(z0 + g).[ ) by RCOMP_1:def_6;
deffunc H1( Element of NAT ) -> Element of REAL = g / ($1 + 2);
consider s1 being Real_Sequence such that
A23: for n being Element of NAT holds s1 . n = H1(n) from SEQ_1:sch_1();
now__::_thesis:_for_n_being_Element_of_NAT_holds_s1_._n_<>_0
let n be Element of NAT ; ::_thesis: s1 . n <> 0
g / (n + 2) <> 0 by A22, XREAL_1:139;
hence s1 . n <> 0 by A23; ::_thesis: verum
end;
then A24: s1 is non-zero by SEQ_1:5;
( s1 is convergent & lim s1 = 0 ) by A23, SEQ_4:31;
then reconsider h = s1 as non-zero 0 -convergent Real_Sequence by A24, FDIFF_1:def_1;
A25: for n being Element of NAT ex z being Real st
( z in N & z in N1 & h . n = z - z0 )
proof
let n be Element of NAT ; ::_thesis: ex z being Real st
( z in N & z in N1 & h . n = z - z0 )
A26: g / (n + 2) > 0 by A22, XREAL_1:139;
0 + 1 < (n + 1) + 1 by XREAL_1:6;
then g / (n + 2) < g / 1 by A22, XREAL_1:76;
then A27: z0 + (g / (n + 2)) < z0 + g by XREAL_1:6;
z0 + (- g) < z0 + (g / (n + 2)) by A22, A26, XREAL_1:6;
then A28: z0 + (g / (n + 2)) in ].(z0 - g),(z0 + g).[ by A27;
take z0 + (g / (n + 2)) ; ::_thesis: ( z0 + (g / (n + 2)) in N & z0 + (g / (n + 2)) in N1 & h . n = (z0 + (g / (n + 2))) - z0 )
thus ( z0 + (g / (n + 2)) in N & z0 + (g / (n + 2)) in N1 & h . n = (z0 + (g / (n + 2))) - z0 ) by A21, A22, A23, A28; ::_thesis: verum
end;
A29: now__::_thesis:_for_n_being_Nat_holds_r_-_s_=_(((h_")_(#)_(R1_/*_h))_-_((h_")_(#)_(R_/*_h)))_._n
let n be Nat; ::_thesis: r - s = (((h ") (#) (R1 /* h)) - ((h ") (#) (R /* h))) . n
A30: n in NAT by ORDINAL1:def_12;
then ex z being Real st
( z in N & z in N1 & h . n = z - z0 ) by A25;
then A31: (r * (h . n)) + (R . (h . n)) = (s * (h . n)) + (R1 . (h . n)) by A19;
A32: h . n <> 0 by A30, SEQ_1:5;
A33: ((r * (h . n)) / (h . n)) + ((R . (h . n)) / (h . n)) = ((s * (h . n)) + (R1 . (h . n))) / (h . n) by A31, XCMPLX_1:62;
A34: (r * (h . n)) / (h . n) = r * ((h . n) / (h . n)) by XCMPLX_1:74
.= r * 1 by A32, XCMPLX_1:60
.= r ;
(s * (h . n)) / (h . n) = s * ((h . n) / (h . n)) by XCMPLX_1:74
.= s * 1 by A32, XCMPLX_1:60
.= s ;
then A35: r + ((R . (h . n)) / (h . n)) = s + ((R1 . (h . n)) / (h . n)) by A33, A34, XCMPLX_1:62;
dom R = REAL by PARTFUN1:def_2;
then A36: rng h c= dom R ;
dom R1 = REAL by PARTFUN1:def_2;
then A37: rng h c= dom R1 ;
A38: (R . (h . n)) / (h . n) = (R . (h . n)) * ((h . n) ") by XCMPLX_0:def_9
.= (R . (h . n)) * ((h ") . n) by VALUED_1:10
.= ((R /* h) . n) * ((h ") . n) by A36, A30, FUNCT_2:108
.= ((h ") (#) (R /* h)) . n by VALUED_1:5 ;
(R1 . (h . n)) / (h . n) = (R1 . (h . n)) * ((h . n) ") by XCMPLX_0:def_9
.= (R1 . (h . n)) * ((h ") . n) by VALUED_1:10
.= ((R1 /* h) . n) * ((h ") . n) by A37, A30, FUNCT_2:108
.= ((h ") (#) (R1 /* h)) . n by VALUED_1:5 ;
then r = s + ((((h ") (#) (R1 /* h)) . n) - (((h ") (#) (R /* h)) . n)) by A35, A38;
hence r - s = (((h ") (#) (R1 /* h)) - ((h ") (#) (R /* h))) . n by A30, RFUNCT_2:1; ::_thesis: verum
end;
then A39: ((h ") (#) (R1 /* h)) - ((h ") (#) (R /* h)) is constant by VALUED_0:def_18;
(((h ") (#) (R1 /* h)) - ((h ") (#) (R /* h))) . 1 = r - s by A29;
then A40: lim (((h ") (#) (R1 /* h)) - ((h ") (#) (R /* h))) = r - s by A39, SEQ_4:25;
A41: ( (h ") (#) (R /* h) is convergent & lim ((h ") (#) (R /* h)) = 0 ) by FDIFF_1:def_2;
( (h ") (#) (R1 /* h) is convergent & lim ((h ") (#) (R1 /* h)) = 0 ) by FDIFF_1:def_2;
then r - s = 0 - 0 by A40, A41, SEQ_2:12;
hence r = s ; ::_thesis: verum
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_differentiable`23_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_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)) ) ) ) );
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
consider x0, y0, z0 being Real such that
A2: ( 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)) ) ) by A1, Def7;
consider N being Neighbourhood of x0 such that
A3: ( 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)) ) by A2;
consider L being LinearFunc, R being RestFunc such that
A4: 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)) by A3;
consider r being Real such that
A5: for p being Real holds L . p = r * p by FDIFF_1:def_3;
take r ; ::_thesis: 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
( r = 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)) ) ) ) )
L . 1 = r * 1 by A5
.= r ;
hence 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
( r = 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)) ) ) ) ) by A2, A3, A4; ::_thesis: verum
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
let r, s be Real; ::_thesis: ( 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
( r = 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
( s = 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)) ) ) ) ) implies r = s )
assume that
A6: 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
( r = 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)) ) ) ) ) and
A7: 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
( s = 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)) ) ) ) ) ; ::_thesis: r = s
consider x0, y0, z0 being Real such that
A8: ( 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
( r = 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)) ) ) ) ) by A6;
consider N being Neighbourhood of x0 such that
A9: ( N c= dom (SVF1 (1,(pdiff1 (f,3)),u)) & ex L being LinearFunc ex R being RestFunc st
( r = 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)) ) ) ) by A8;
consider L being LinearFunc, R being RestFunc such that
A10: ( r = 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)) ) ) by A9;
consider x1, y1, z1 being Real such that
A11: ( u = <*x1,y1,z1*> & ex N being Neighbourhood of x1 st
( N c= dom (SVF1 (1,(pdiff1 (f,3)),u)) & ex L being LinearFunc ex R being RestFunc st
( s = L . 1 & ( for x being Real st x in N holds
((SVF1 (1,(pdiff1 (f,3)),u)) . x) - ((SVF1 (1,(pdiff1 (f,3)),u)) . x1) = (L . (x - x1)) + (R . (x - x1)) ) ) ) ) by A7;
consider N1 being Neighbourhood of x1 such that
A12: ( N1 c= dom (SVF1 (1,(pdiff1 (f,3)),u)) & ex L being LinearFunc ex R being RestFunc st
( s = L . 1 & ( for x being Real st x in N1 holds
((SVF1 (1,(pdiff1 (f,3)),u)) . x) - ((SVF1 (1,(pdiff1 (f,3)),u)) . x1) = (L . (x - x1)) + (R . (x - x1)) ) ) ) by A11;
consider L1 being LinearFunc, R1 being RestFunc such that
A13: ( s = L1 . 1 & ( for x being Real st x in N1 holds
((SVF1 (1,(pdiff1 (f,3)),u)) . x) - ((SVF1 (1,(pdiff1 (f,3)),u)) . x1) = (L1 . (x - x1)) + (R1 . (x - x1)) ) ) by A12;
consider r1 being Real such that
A14: for p being Real holds L . p = r1 * p by FDIFF_1:def_3;
consider p1 being Real such that
A15: for p being Real holds L1 . p = p1 * p by FDIFF_1:def_3;
A16: r = r1 * 1 by A10, A14;
A17: s = p1 * 1 by A13, A15;
A18: ( x0 = x1 & y0 = y1 & z0 = z1 ) by A8, A11, FINSEQ_1:78;
A19: now__::_thesis:_for_x_being_Real_st_x_in_N_&_x_in_N1_holds_
(r_*_(x_-_x0))_+_(R_._(x_-_x0))_=_(s_*_(x_-_x0))_+_(R1_._(x_-_x0))
let x be Real; ::_thesis: ( x in N & x in N1 implies (r * (x - x0)) + (R . (x - x0)) = (s * (x - x0)) + (R1 . (x - x0)) )
assume A20: ( x in N & x in N1 ) ; ::_thesis: (r * (x - x0)) + (R . (x - x0)) = (s * (x - x0)) + (R1 . (x - x0))
then ((SVF1 (1,(pdiff1 (f,3)),u)) . x) - ((SVF1 (1,(pdiff1 (f,3)),u)) . x0) = (L . (x - x0)) + (R . (x - x0)) by A10;
then (L . (x - x0)) + (R . (x - x0)) = (L1 . (x - x0)) + (R1 . (x - x0)) by A13, A18, A20;
then (r1 * (x - x0)) + (R . (x - x0)) = (L1 . (x - x0)) + (R1 . (x - x0)) by A14;
hence (r * (x - x0)) + (R . (x - x0)) = (s * (x - x0)) + (R1 . (x - x0)) by A15, A16, A17; ::_thesis: verum
end;
consider N0 being Neighbourhood of x0 such that
A21: ( N0 c= N & N0 c= N1 ) by A18, RCOMP_1:17;
consider g being real number such that
A22: ( 0 < g & N0 = ].(x0 - g),(x0 + g).[ ) by RCOMP_1:def_6;
deffunc H1( Element of NAT ) -> Element of REAL = g / ($1 + 2);
consider s1 being Real_Sequence such that
A23: for n being Element of NAT holds s1 . n = H1(n) from SEQ_1:sch_1();
now__::_thesis:_for_n_being_Element_of_NAT_holds_s1_._n_<>_0
let n be Element of NAT ; ::_thesis: s1 . n <> 0
g / (n + 2) <> 0 by A22, XREAL_1:139;
hence s1 . n <> 0 by A23; ::_thesis: verum
end;
then A24: s1 is non-zero by SEQ_1:5;
( s1 is convergent & lim s1 = 0 ) by A23, SEQ_4:31;
then reconsider h = s1 as non-zero 0 -convergent Real_Sequence by A24, FDIFF_1:def_1;
A25: for n being Element of NAT ex x being Real st
( x in N & x in N1 & h . n = x - x0 )
proof
let n be Element of NAT ; ::_thesis: ex x being Real st
( x in N & x in N1 & h . n = x - x0 )
A26: g / (n + 2) > 0 by A22, XREAL_1:139;
0 + 1 < (n + 1) + 1 by XREAL_1:6;
then g / (n + 2) < g / 1 by A22, XREAL_1:76;
then A27: x0 + (g / (n + 2)) < x0 + g by XREAL_1:6;
x0 + (- g) < x0 + (g / (n + 2)) by A22, A26, XREAL_1:6;
then A28: x0 + (g / (n + 2)) in ].(x0 - g),(x0 + g).[ by A27;
take x0 + (g / (n + 2)) ; ::_thesis: ( x0 + (g / (n + 2)) in N & x0 + (g / (n + 2)) in N1 & h . n = (x0 + (g / (n + 2))) - x0 )
thus ( x0 + (g / (n + 2)) in N & x0 + (g / (n + 2)) in N1 & h . n = (x0 + (g / (n + 2))) - x0 ) by A21, A22, A23, A28; ::_thesis: verum
end;
A29: now__::_thesis:_for_n_being_Nat_holds_r_-_s_=_(((h_")_(#)_(R1_/*_h))_-_((h_")_(#)_(R_/*_h)))_._n
let n be Nat; ::_thesis: r - s = (((h ") (#) (R1 /* h)) - ((h ") (#) (R /* h))) . n
A30: n in NAT by ORDINAL1:def_12;
then ex x being Real st
( x in N & x in N1 & h . n = x - x0 ) by A25;
then A31: (r * (h . n)) + (R . (h . n)) = (s * (h . n)) + (R1 . (h . n)) by A19;
A32: h . n <> 0 by A30, SEQ_1:5;
A33: ((r * (h . n)) / (h . n)) + ((R . (h . n)) / (h . n)) = ((s * (h . n)) + (R1 . (h . n))) / (h . n) by A31, XCMPLX_1:62;
A34: (r * (h . n)) / (h . n) = r * ((h . n) / (h . n)) by XCMPLX_1:74
.= r * 1 by A32, XCMPLX_1:60
.= r ;
(s * (h . n)) / (h . n) = s * ((h . n) / (h . n)) by XCMPLX_1:74
.= s * 1 by A32, XCMPLX_1:60
.= s ;
then A35: r + ((R . (h . n)) / (h . n)) = s + ((R1 . (h . n)) / (h . n)) by A33, A34, XCMPLX_1:62;
dom R = REAL by PARTFUN1:def_2;
then A36: rng h c= dom R ;
dom R1 = REAL by PARTFUN1:def_2;
then A37: rng h c= dom R1 ;
A38: (R . (h . n)) / (h . n) = (R . (h . n)) * ((h . n) ") by XCMPLX_0:def_9
.= (R . (h . n)) * ((h ") . n) by VALUED_1:10
.= ((R /* h) . n) * ((h ") . n) by A36, A30, FUNCT_2:108
.= ((h ") (#) (R /* h)) . n by VALUED_1:5 ;
(R1 . (h . n)) / (h . n) = (R1 . (h . n)) * ((h . n) ") by XCMPLX_0:def_9
.= (R1 . (h . n)) * ((h ") . n) by VALUED_1:10
.= ((R1 /* h) . n) * ((h ") . n) by A37, A30, FUNCT_2:108
.= ((h ") (#) (R1 /* h)) . n by VALUED_1:5 ;
then r = s + ((((h ") (#) (R1 /* h)) . n) - (((h ") (#) (R /* h)) . n)) by A35, A38;
hence r - s = (((h ") (#) (R1 /* h)) - ((h ") (#) (R /* h))) . n by A30, VALUED_1:15; ::_thesis: verum
end;
then A39: ((h ") (#) (R1 /* h)) - ((h ") (#) (R /* h)) is constant by VALUED_0:def_18;
(((h ") (#) (R1 /* h)) - ((h ") (#) (R /* h))) . 1 = r - s by A29;
then A40: lim (((h ") (#) (R1 /* h)) - ((h ") (#) (R /* h))) = r - s by A39, SEQ_4:25;
A41: ( (h ") (#) (R /* h) is convergent & lim ((h ") (#) (R /* h)) = 0 ) by FDIFF_1:def_2;
( (h ") (#) (R1 /* h) is convergent & lim ((h ") (#) (R1 /* h)) = 0 ) by FDIFF_1:def_2;
then r - s = 0 - 0 by A40, A41, SEQ_2:12;
hence r = s ; ::_thesis: verum
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_differentiable`31_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_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)) ) ) ) );
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
consider x0, y0, z0 being Real such that
A2: ( 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)) ) ) by A1, Def8;
consider N being Neighbourhood of y0 such that
A3: ( 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)) ) by A2;
consider L being LinearFunc, R being RestFunc such that
A4: 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)) by A3;
consider r being Real such that
A5: for p being Real holds L . p = r * p by FDIFF_1:def_3;
take r ; ::_thesis: 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
( r = 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)) ) ) ) )
L . 1 = r * 1 by A5
.= r ;
hence 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
( r = 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)) ) ) ) ) by A2, A3, A4; ::_thesis: verum
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
let r, s be Real; ::_thesis: ( 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
( r = 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
( s = 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)) ) ) ) ) implies r = s )
assume that
A6: 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
( r = 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)) ) ) ) ) and
A7: 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
( s = 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)) ) ) ) ) ; ::_thesis: r = s
consider x0, y0, z0 being Real such that
A8: ( 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
( r = 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)) ) ) ) ) by A6;
consider N being Neighbourhood of y0 such that
A9: ( N c= dom (SVF1 (2,(pdiff1 (f,3)),u)) & ex L being LinearFunc ex R being RestFunc st
( r = 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)) ) ) ) by A8;
consider L being LinearFunc, R being RestFunc such that
A10: ( r = 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)) ) ) by A9;
consider x1, y1, z1 being Real such that
A11: ( u = <*x1,y1,z1*> & ex N being Neighbourhood of y1 st
( N c= dom (SVF1 (2,(pdiff1 (f,3)),u)) & ex L being LinearFunc ex R being RestFunc st
( s = L . 1 & ( for y being Real st y in N holds
((SVF1 (2,(pdiff1 (f,3)),u)) . y) - ((SVF1 (2,(pdiff1 (f,3)),u)) . y1) = (L . (y - y1)) + (R . (y - y1)) ) ) ) ) by A7;
consider N1 being Neighbourhood of y1 such that
A12: ( N1 c= dom (SVF1 (2,(pdiff1 (f,3)),u)) & ex L being LinearFunc ex R being RestFunc st
( s = L . 1 & ( for y being Real st y in N1 holds
((SVF1 (2,(pdiff1 (f,3)),u)) . y) - ((SVF1 (2,(pdiff1 (f,3)),u)) . y1) = (L . (y - y1)) + (R . (y - y1)) ) ) ) by A11;
consider L1 being LinearFunc, R1 being RestFunc such that
A13: ( s = L1 . 1 & ( for y being Real st y in N1 holds
((SVF1 (2,(pdiff1 (f,3)),u)) . y) - ((SVF1 (2,(pdiff1 (f,3)),u)) . y1) = (L1 . (y - y1)) + (R1 . (y - y1)) ) ) by A12;
consider r1 being Real such that
A14: for p being Real holds L . p = r1 * p by FDIFF_1:def_3;
consider p1 being Real such that
A15: for p being Real holds L1 . p = p1 * p by FDIFF_1:def_3;
A16: r = r1 * 1 by A10, A14;
A17: s = p1 * 1 by A13, A15;
A18: ( x0 = x1 & y0 = y1 & z0 = z1 ) by A8, A11, FINSEQ_1:78;
A19: now__::_thesis:_for_y_being_Real_st_y_in_N_&_y_in_N1_holds_
(r_*_(y_-_y0))_+_(R_._(y_-_y0))_=_(s_*_(y_-_y0))_+_(R1_._(y_-_y0))
let y be Real; ::_thesis: ( y in N & y in N1 implies (r * (y - y0)) + (R . (y - y0)) = (s * (y - y0)) + (R1 . (y - y0)) )
assume A20: ( y in N & y in N1 ) ; ::_thesis: (r * (y - y0)) + (R . (y - y0)) = (s * (y - y0)) + (R1 . (y - y0))
then ((SVF1 (2,(pdiff1 (f,3)),u)) . y) - ((SVF1 (2,(pdiff1 (f,3)),u)) . y0) = (L . (y - y0)) + (R . (y - y0)) by A10;
then (L . (y - y0)) + (R . (y - y0)) = (L1 . (y - y0)) + (R1 . (y - y0)) by A13, A18, A20;
then (r1 * (y - y0)) + (R . (y - y0)) = (L1 . (y - y0)) + (R1 . (y - y0)) by A14;
hence (r * (y - y0)) + (R . (y - y0)) = (s * (y - y0)) + (R1 . (y - y0)) by A15, A16, A17; ::_thesis: verum
end;
consider N0 being Neighbourhood of y0 such that
A21: ( N0 c= N & N0 c= N1 ) by A18, RCOMP_1:17;
consider g being real number such that
A22: ( 0 < g & N0 = ].(y0 - g),(y0 + g).[ ) by RCOMP_1:def_6;
deffunc H1( Element of NAT ) -> Element of REAL = g / ($1 + 2);
consider s1 being Real_Sequence such that
A23: for n being Element of NAT holds s1 . n = H1(n) from SEQ_1:sch_1();
now__::_thesis:_for_n_being_Element_of_NAT_holds_s1_._n_<>_0
let n be Element of NAT ; ::_thesis: s1 . n <> 0
g / (n + 2) <> 0 by A22, XREAL_1:139;
hence s1 . n <> 0 by A23; ::_thesis: verum
end;
then A24: s1 is non-zero by SEQ_1:5;
( s1 is convergent & lim s1 = 0 ) by A23, SEQ_4:31;
then reconsider h = s1 as non-zero 0 -convergent Real_Sequence by A24, FDIFF_1:def_1;
A25: for n being Element of NAT ex y being Real st
( y in N & y in N1 & h . n = y - y0 )
proof
let n be Element of NAT ; ::_thesis: ex y being Real st
( y in N & y in N1 & h . n = y - y0 )
A26: g / (n + 2) > 0 by A22, XREAL_1:139;
0 + 1 < (n + 1) + 1 by XREAL_1:6;
then g / (n + 2) < g / 1 by A22, XREAL_1:76;
then A27: y0 + (g / (n + 2)) < y0 + g by XREAL_1:6;
y0 + (- g) < y0 + (g / (n + 2)) by A22, A26, XREAL_1:6;
then A28: y0 + (g / (n + 2)) in ].(y0 - g),(y0 + g).[ by A27;
take y0 + (g / (n + 2)) ; ::_thesis: ( y0 + (g / (n + 2)) in N & y0 + (g / (n + 2)) in N1 & h . n = (y0 + (g / (n + 2))) - y0 )
thus ( y0 + (g / (n + 2)) in N & y0 + (g / (n + 2)) in N1 & h . n = (y0 + (g / (n + 2))) - y0 ) by A21, A22, A23, A28; ::_thesis: verum
end;
A29: now__::_thesis:_for_n_being_Nat_holds_r_-_s_=_(((h_")_(#)_(R1_/*_h))_-_((h_")_(#)_(R_/*_h)))_._n
let n be Nat; ::_thesis: r - s = (((h ") (#) (R1 /* h)) - ((h ") (#) (R /* h))) . n
A30: n in NAT by ORDINAL1:def_12;
then ex y being Real st
( y in N & y in N1 & h . n = y - y0 ) by A25;
then A31: (r * (h . n)) + (R . (h . n)) = (s * (h . n)) + (R1 . (h . n)) by A19;
A32: h . n <> 0 by A30, SEQ_1:5;
A33: ((r * (h . n)) / (h . n)) + ((R . (h . n)) / (h . n)) = ((s * (h . n)) + (R1 . (h . n))) / (h . n) by A31, XCMPLX_1:62;
A34: (r * (h . n)) / (h . n) = r * ((h . n) / (h . n)) by XCMPLX_1:74
.= r * 1 by A32, XCMPLX_1:60
.= r ;
(s * (h . n)) / (h . n) = s * ((h . n) / (h . n)) by XCMPLX_1:74
.= s * 1 by A32, XCMPLX_1:60
.= s ;
then A35: r + ((R . (h . n)) / (h . n)) = s + ((R1 . (h . n)) / (h . n)) by A33, A34, XCMPLX_1:62;
dom R = REAL by PARTFUN1:def_2;
then A36: rng h c= dom R ;
dom R1 = REAL by PARTFUN1:def_2;
then A37: rng h c= dom R1 ;
A38: (R . (h . n)) / (h . n) = (R . (h . n)) * ((h . n) ") by XCMPLX_0:def_9
.= (R . (h . n)) * ((h ") . n) by VALUED_1:10
.= ((R /* h) . n) * ((h ") . n) by A36, A30, FUNCT_2:108
.= ((h ") (#) (R /* h)) . n by VALUED_1:5 ;
(R1 . (h . n)) / (h . n) = (R1 . (h . n)) * ((h . n) ") by XCMPLX_0:def_9
.= (R1 . (h . n)) * ((h ") . n) by VALUED_1:10
.= ((R1 /* h) . n) * ((h ") . n) by A37, A30, FUNCT_2:108
.= ((h ") (#) (R1 /* h)) . n by VALUED_1:5 ;
then r = s + ((((h ") (#) (R1 /* h)) . n) - (((h ") (#) (R /* h)) . n)) by A35, A38;
hence r - s = (((h ") (#) (R1 /* h)) - ((h ") (#) (R /* h))) . n by A30, RFUNCT_2:1; ::_thesis: verum
end;
then A39: ((h ") (#) (R1 /* h)) - ((h ") (#) (R /* h)) is constant by VALUED_0:def_18;
(((h ") (#) (R1 /* h)) - ((h ") (#) (R /* h))) . 1 = r - s by A29;
then A40: lim (((h ") (#) (R1 /* h)) - ((h ") (#) (R /* h))) = r - s by A39, SEQ_4:25;
A41: ( (h ") (#) (R /* h) is convergent & lim ((h ") (#) (R /* h)) = 0 ) by FDIFF_1:def_2;
( (h ") (#) (R1 /* h) is convergent & lim ((h ") (#) (R1 /* h)) = 0 ) by FDIFF_1:def_2;
then r - s = 0 - 0 by A40, A41, SEQ_2:12;
hence r = s ; ::_thesis: verum
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_differentiable`32_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_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)) ) ) ) );
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
consider x0, y0, z0 being Real such that
A2: ( 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)) ) ) by A1, Def9;
consider N being Neighbourhood of z0 such that
A3: ( 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)) ) by A2;
consider L being LinearFunc, R being RestFunc such that
A4: 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)) by A3;
consider r being Real such that
A5: for p being Real holds L . p = r * p by FDIFF_1:def_3;
take r ; ::_thesis: 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
( r = 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)) ) ) ) )
L . 1 = r * 1 by A5
.= r ;
hence 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
( r = 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)) ) ) ) ) by A2, A3, A4; ::_thesis: verum
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
let r, s be Real; ::_thesis: ( 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
( r = 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
( s = 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)) ) ) ) ) implies r = s )
assume that
A6: 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
( r = 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)) ) ) ) ) and
A7: 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
( s = 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)) ) ) ) ) ; ::_thesis: r = s
consider x0, y0, z0 being Real such that
A8: ( 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
( r = 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)) ) ) ) ) by A6;
consider N being Neighbourhood of z0 such that
A9: ( N c= dom (SVF1 (3,(pdiff1 (f,3)),u)) & ex L being LinearFunc ex R being RestFunc st
( r = 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)) ) ) ) by A8;
consider L being LinearFunc, R being RestFunc such that
A10: ( r = 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)) ) ) by A9;
consider x1, y1, z1 being Real such that
A11: ( u = <*x1,y1,z1*> & ex N being Neighbourhood of z1 st
( N c= dom (SVF1 (3,(pdiff1 (f,3)),u)) & ex L being LinearFunc ex R being RestFunc st
( s = L . 1 & ( for z being Real st z in N holds
((SVF1 (3,(pdiff1 (f,3)),u)) . z) - ((SVF1 (3,(pdiff1 (f,3)),u)) . z1) = (L . (z - z1)) + (R . (z - z1)) ) ) ) ) by A7;
consider N1 being Neighbourhood of z1 such that
A12: ( N1 c= dom (SVF1 (3,(pdiff1 (f,3)),u)) & ex L being LinearFunc ex R being RestFunc st
( s = L . 1 & ( for z being Real st z in N1 holds
((SVF1 (3,(pdiff1 (f,3)),u)) . z) - ((SVF1 (3,(pdiff1 (f,3)),u)) . z1) = (L . (z - z1)) + (R . (z - z1)) ) ) ) by A11;
consider L1 being LinearFunc, R1 being RestFunc such that
A13: ( s = L1 . 1 & ( for z being Real st z in N1 holds
((SVF1 (3,(pdiff1 (f,3)),u)) . z) - ((SVF1 (3,(pdiff1 (f,3)),u)) . z1) = (L1 . (z - z1)) + (R1 . (z - z1)) ) ) by A12;
consider r1 being Real such that
A14: for p being Real holds L . p = r1 * p by FDIFF_1:def_3;
consider p1 being Real such that
A15: for p being Real holds L1 . p = p1 * p by FDIFF_1:def_3;
A16: r = r1 * 1 by A10, A14;
A17: s = p1 * 1 by A13, A15;
A18: ( x0 = x1 & y0 = y1 & z0 = z1 ) by A8, A11, FINSEQ_1:78;
A19: now__::_thesis:_for_z_being_Real_st_z_in_N_&_z_in_N1_holds_
(r_*_(z_-_z0))_+_(R_._(z_-_z0))_=_(s_*_(z_-_z0))_+_(R1_._(z_-_z0))
let z be Real; ::_thesis: ( z in N & z in N1 implies (r * (z - z0)) + (R . (z - z0)) = (s * (z - z0)) + (R1 . (z - z0)) )
assume A20: ( z in N & z in N1 ) ; ::_thesis: (r * (z - z0)) + (R . (z - z0)) = (s * (z - z0)) + (R1 . (z - z0))
then ((SVF1 (3,(pdiff1 (f,3)),u)) . z) - ((SVF1 (3,(pdiff1 (f,3)),u)) . z0) = (L . (z - z0)) + (R . (z - z0)) by A10;
then (L . (z - z0)) + (R . (z - z0)) = (L1 . (z - z0)) + (R1 . (z - z0)) by A13, A18, A20;
then (r1 * (z - z0)) + (R . (z - z0)) = (L1 . (z - z0)) + (R1 . (z - z0)) by A14;
hence (r * (z - z0)) + (R . (z - z0)) = (s * (z - z0)) + (R1 . (z - z0)) by A15, A16, A17; ::_thesis: verum
end;
consider N0 being Neighbourhood of z0 such that
A21: ( N0 c= N & N0 c= N1 ) by A18, RCOMP_1:17;
consider g being real number such that
A22: ( 0 < g & N0 = ].(z0 - g),(z0 + g).[ ) by RCOMP_1:def_6;
deffunc H1( Element of NAT ) -> Element of REAL = g / ($1 + 2);
consider s1 being Real_Sequence such that
A23: for n being Element of NAT holds s1 . n = H1(n) from SEQ_1:sch_1();
now__::_thesis:_for_n_being_Element_of_NAT_holds_s1_._n_<>_0
let n be Element of NAT ; ::_thesis: s1 . n <> 0
g / (n + 2) <> 0 by A22, XREAL_1:139;
hence s1 . n <> 0 by A23; ::_thesis: verum
end;
then A24: s1 is non-zero by SEQ_1:5;
( s1 is convergent & lim s1 = 0 ) by A23, SEQ_4:31;
then reconsider h = s1 as non-zero 0 -convergent Real_Sequence by A24, FDIFF_1:def_1;
A25: for n being Element of NAT ex z being Real st
( z in N & z in N1 & h . n = z - z0 )
proof
let n be Element of NAT ; ::_thesis: ex z being Real st
( z in N & z in N1 & h . n = z - z0 )
A26: g / (n + 2) > 0 by A22, XREAL_1:139;
0 + 1 < (n + 1) + 1 by XREAL_1:6;
then g / (n + 2) < g / 1 by A22, XREAL_1:76;
then A27: z0 + (g / (n + 2)) < z0 + g by XREAL_1:6;
z0 + (- g) < z0 + (g / (n + 2)) by A22, A26, XREAL_1:6;
then A28: z0 + (g / (n + 2)) in ].(z0 - g),(z0 + g).[ by A27;
take z0 + (g / (n + 2)) ; ::_thesis: ( z0 + (g / (n + 2)) in N & z0 + (g / (n + 2)) in N1 & h . n = (z0 + (g / (n + 2))) - z0 )
thus ( z0 + (g / (n + 2)) in N & z0 + (g / (n + 2)) in N1 & h . n = (z0 + (g / (n + 2))) - z0 ) by A21, A22, A23, A28; ::_thesis: verum
end;
A29: now__::_thesis:_for_n_being_Nat_holds_r_-_s_=_(((h_")_(#)_(R1_/*_h))_-_((h_")_(#)_(R_/*_h)))_._n
let n be Nat; ::_thesis: r - s = (((h ") (#) (R1 /* h)) - ((h ") (#) (R /* h))) . n
A30: n in NAT by ORDINAL1:def_12;
then ex z being Real st
( z in N & z in N1 & h . n = z - z0 ) by A25;
then A31: (r * (h . n)) + (R . (h . n)) = (s * (h . n)) + (R1 . (h . n)) by A19;
A32: h . n <> 0 by A30, SEQ_1:5;
A33: ((r * (h . n)) / (h . n)) + ((R . (h . n)) / (h . n)) = ((s * (h . n)) + (R1 . (h . n))) / (h . n) by A31, XCMPLX_1:62;
A34: (r * (h . n)) / (h . n) = r * ((h . n) / (h . n)) by XCMPLX_1:74
.= r * 1 by A32, XCMPLX_1:60
.= r ;
(s * (h . n)) / (h . n) = s * ((h . n) / (h . n)) by XCMPLX_1:74
.= s * 1 by A32, XCMPLX_1:60
.= s ;
then A35: r + ((R . (h . n)) / (h . n)) = s + ((R1 . (h . n)) / (h . n)) by A33, A34, XCMPLX_1:62;
dom R = REAL by PARTFUN1:def_2;
then A36: rng h c= dom R ;
dom R1 = REAL by PARTFUN1:def_2;
then A37: rng h c= dom R1 ;
A38: (R . (h . n)) / (h . n) = (R . (h . n)) * ((h . n) ") by XCMPLX_0:def_9
.= (R . (h . n)) * ((h ") . n) by VALUED_1:10
.= ((R /* h) . n) * ((h ") . n) by A36, A30, FUNCT_2:108
.= ((h ") (#) (R /* h)) . n by VALUED_1:5 ;
(R1 . (h . n)) / (h . n) = (R1 . (h . n)) * ((h . n) ") by XCMPLX_0:def_9
.= (R1 . (h . n)) * ((h ") . n) by VALUED_1:10
.= ((R1 /* h) . n) * ((h ") . n) by A37, A30, FUNCT_2:108
.= ((h ") (#) (R1 /* h)) . n by VALUED_1:5 ;
then r = s + ((((h ") (#) (R1 /* h)) . n) - (((h ") (#) (R /* h)) . n)) by A35, A38;
hence r - s = (((h ") (#) (R1 /* h)) - ((h ") (#) (R /* h))) . n by A30, RFUNCT_2:1; ::_thesis: verum
end;
then A39: ((h ") (#) (R1 /* h)) - ((h ") (#) (R /* h)) is constant by VALUED_0:def_18;
(((h ") (#) (R1 /* h)) - ((h ") (#) (R /* h))) . 1 = r - s by A29;
then A40: lim (((h ") (#) (R1 /* h)) - ((h ") (#) (R /* h))) = r - s by A39, SEQ_4:25;
A41: ( (h ") (#) (R /* h) is convergent & lim ((h ") (#) (R /* h)) = 0 ) by FDIFF_1:def_2;
( (h ") (#) (R1 /* h) is convergent & lim ((h ") (#) (R1 /* h)) = 0 ) by FDIFF_1:def_2;
then r - s = 0 - 0 by A40, A41, SEQ_2:12;
hence r = s ; ::_thesis: verum
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_differentiable`33_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_differentiable`11_in u holds
SVF1 (1,(pdiff1 (f,1)),u) is_differentiable_in x0
proof
let x0, y0, z0 be Real; ::_thesis: 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
let u be Element of REAL 3; ::_thesis: 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
let f be PartFunc of (REAL 3),REAL; ::_thesis: ( u = <*x0,y0,z0*> & f is_hpartial_differentiable`11_in u implies SVF1 (1,(pdiff1 (f,1)),u) is_differentiable_in x0 )
assume that
A1: u = <*x0,y0,z0*> and
A2: f is_hpartial_differentiable`11_in u ; ::_thesis: SVF1 (1,(pdiff1 (f,1)),u) is_differentiable_in x0
consider x1, y1, z1 being Real such that
A3: ( u = <*x1,y1,z1*> & ex N being Neighbourhood of x1 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)) . x1) = (L . (x - x1)) + (R . (x - x1)) ) ) by A2, Def1;
x0 = x1 by A1, A3, FINSEQ_1:78;
hence SVF1 (1,(pdiff1 (f,1)),u) is_differentiable_in x0 by A3, FDIFF_1:def_4; ::_thesis: verum
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
let x0, y0, z0 be Real; ::_thesis: 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
let u be Element of REAL 3; ::_thesis: 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
let f be PartFunc of (REAL 3),REAL; ::_thesis: ( u = <*x0,y0,z0*> & f is_hpartial_differentiable`12_in u implies SVF1 (2,(pdiff1 (f,1)),u) is_differentiable_in y0 )
assume that
A1: u = <*x0,y0,z0*> and
A2: f is_hpartial_differentiable`12_in u ; ::_thesis: SVF1 (2,(pdiff1 (f,1)),u) is_differentiable_in y0
consider x1, y1, z1 being Real such that
A3: ( u = <*x1,y1,z1*> & ex N being Neighbourhood of y1 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)) . y1) = (L . (y - y1)) + (R . (y - y1)) ) ) by A2, Def2;
y0 = y1 by A1, A3, FINSEQ_1:78;
hence SVF1 (2,(pdiff1 (f,1)),u) is_differentiable_in y0 by A3, FDIFF_1:def_4; ::_thesis: verum
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
let x0, y0, z0 be Real; ::_thesis: 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
let u be Element of REAL 3; ::_thesis: 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
let f be PartFunc of (REAL 3),REAL; ::_thesis: ( u = <*x0,y0,z0*> & f is_hpartial_differentiable`13_in u implies SVF1 (3,(pdiff1 (f,1)),u) is_differentiable_in z0 )
assume that
A1: u = <*x0,y0,z0*> and
A2: f is_hpartial_differentiable`13_in u ; ::_thesis: SVF1 (3,(pdiff1 (f,1)),u) is_differentiable_in z0
consider x1, y1, z1 being Real such that
A3: ( u = <*x1,y1,z1*> & ex N being Neighbourhood of z1 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)) . z1) = (L . (z - z1)) + (R . (z - z1)) ) ) by A2, Def3;
z0 = z1 by A1, A3, FINSEQ_1:78;
hence SVF1 (3,(pdiff1 (f,1)),u) is_differentiable_in z0 by A3, FDIFF_1:def_4; ::_thesis: verum
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
let x0, y0, z0 be Real; ::_thesis: 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
let u be Element of REAL 3; ::_thesis: 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
let f be PartFunc of (REAL 3),REAL; ::_thesis: ( u = <*x0,y0,z0*> & f is_hpartial_differentiable`21_in u implies SVF1 (1,(pdiff1 (f,2)),u) is_differentiable_in x0 )
assume that
A1: u = <*x0,y0,z0*> and
A2: f is_hpartial_differentiable`21_in u ; ::_thesis: SVF1 (1,(pdiff1 (f,2)),u) is_differentiable_in x0
consider x1, y1, z1 being Real such that
A3: ( u = <*x1,y1,z1*> & ex N being Neighbourhood of x1 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)) . x1) = (L . (x - x1)) + (R . (x - x1)) ) ) by A2, Def4;
x0 = x1 by A1, A3, FINSEQ_1:78;
hence SVF1 (1,(pdiff1 (f,2)),u) is_differentiable_in x0 by A3, FDIFF_1:def_4; ::_thesis: verum
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
let x0, y0, z0 be Real; ::_thesis: 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
let u be Element of REAL 3; ::_thesis: 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
let f be PartFunc of (REAL 3),REAL; ::_thesis: ( u = <*x0,y0,z0*> & f is_hpartial_differentiable`22_in u implies SVF1 (2,(pdiff1 (f,2)),u) is_differentiable_in y0 )
assume that
A1: u = <*x0,y0,z0*> and
A2: f is_hpartial_differentiable`22_in u ; ::_thesis: SVF1 (2,(pdiff1 (f,2)),u) is_differentiable_in y0
consider x1, y1, z1 being Real such that
A3: ( u = <*x1,y1,z1*> & ex N being Neighbourhood of y1 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)) . y1) = (L . (y - y1)) + (R . (y - y1)) ) ) by A2, Def5;
y0 = y1 by A1, A3, FINSEQ_1:78;
hence SVF1 (2,(pdiff1 (f,2)),u) is_differentiable_in y0 by A3, FDIFF_1:def_4; ::_thesis: verum
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
let x0, y0, z0 be Real; ::_thesis: 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
let u be Element of REAL 3; ::_thesis: 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
let f be PartFunc of (REAL 3),REAL; ::_thesis: ( u = <*x0,y0,z0*> & f is_hpartial_differentiable`23_in u implies SVF1 (3,(pdiff1 (f,2)),u) is_differentiable_in z0 )
assume that
A1: u = <*x0,y0,z0*> and
A2: f is_hpartial_differentiable`23_in u ; ::_thesis: SVF1 (3,(pdiff1 (f,2)),u) is_differentiable_in z0
consider x1, y1, z1 being Real such that
A3: ( u = <*x1,y1,z1*> & ex N being Neighbourhood of z1 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)) . z1) = (L . (z - z1)) + (R . (z - z1)) ) ) by A2, Def6;
z0 = z1 by A1, A3, FINSEQ_1:78;
hence SVF1 (3,(pdiff1 (f,2)),u) is_differentiable_in z0 by A3, FDIFF_1:def_4; ::_thesis: verum
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
let x0, y0, z0 be Real; ::_thesis: 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
let u be Element of REAL 3; ::_thesis: 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
let f be PartFunc of (REAL 3),REAL; ::_thesis: ( u = <*x0,y0,z0*> & f is_hpartial_differentiable`31_in u implies SVF1 (1,(pdiff1 (f,3)),u) is_differentiable_in x0 )
assume that
A1: u = <*x0,y0,z0*> and
A2: f is_hpartial_differentiable`31_in u ; ::_thesis: SVF1 (1,(pdiff1 (f,3)),u) is_differentiable_in x0
consider x1, y1, z1 being Real such that
A3: ( u = <*x1,y1,z1*> & ex N being Neighbourhood of x1 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)) . x1) = (L . (x - x1)) + (R . (x - x1)) ) ) by A2, Def7;
x0 = x1 by A1, A3, FINSEQ_1:78;
hence SVF1 (1,(pdiff1 (f,3)),u) is_differentiable_in x0 by A3, FDIFF_1:def_4; ::_thesis: verum
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
let x0, y0, z0 be Real; ::_thesis: 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
let u be Element of REAL 3; ::_thesis: 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
let f be PartFunc of (REAL 3),REAL; ::_thesis: ( u = <*x0,y0,z0*> & f is_hpartial_differentiable`32_in u implies SVF1 (2,(pdiff1 (f,3)),u) is_differentiable_in y0 )
assume that
A1: u = <*x0,y0,z0*> and
A2: f is_hpartial_differentiable`32_in u ; ::_thesis: SVF1 (2,(pdiff1 (f,3)),u) is_differentiable_in y0
consider x1, y1, z1 being Real such that
A3: ( u = <*x1,y1,z1*> & ex N being Neighbourhood of y1 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)) . y1) = (L . (y - y1)) + (R . (y - y1)) ) ) by A2, Def8;
y0 = y1 by A1, A3, FINSEQ_1:78;
hence SVF1 (2,(pdiff1 (f,3)),u) is_differentiable_in y0 by A3, FDIFF_1:def_4; ::_thesis: verum
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
let x0, y0, z0 be Real; ::_thesis: 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
let u be Element of REAL 3; ::_thesis: 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
let f be PartFunc of (REAL 3),REAL; ::_thesis: ( u = <*x0,y0,z0*> & f is_hpartial_differentiable`33_in u implies SVF1 (3,(pdiff1 (f,3)),u) is_differentiable_in z0 )
assume that
A1: u = <*x0,y0,z0*> and
A2: f is_hpartial_differentiable`33_in u ; ::_thesis: SVF1 (3,(pdiff1 (f,3)),u) is_differentiable_in z0
consider x1, y1, z1 being Real such that
A3: ( u = <*x1,y1,z1*> & ex N being Neighbourhood of z1 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)) . z1) = (L . (z - z1)) + (R . (z - z1)) ) ) by A2, Def9;
z0 = z1 by A1, A3, FINSEQ_1:78;
hence SVF1 (3,(pdiff1 (f,3)),u) is_differentiable_in z0 by A3, FDIFF_1:def_4; ::_thesis: verum
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
let x0, y0, z0 be Real; ::_thesis: 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)
let u be Element of REAL 3; ::_thesis: 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)
let f be PartFunc of (REAL 3),REAL; ::_thesis: ( u = <*x0,y0,z0*> & f is_hpartial_differentiable`11_in u implies hpartdiff11 (f,u) = diff ((SVF1 (1,(pdiff1 (f,1)),u)),x0) )
set r = hpartdiff11 (f,u);
assume that
A1: u = <*x0,y0,z0*> and
A2: f is_hpartial_differentiable`11_in u ; ::_thesis: hpartdiff11 (f,u) = diff ((SVF1 (1,(pdiff1 (f,1)),u)),x0)
consider x1, y1, z1 being Real such that
A3: ( u = <*x1,y1,z1*> & ex N being Neighbourhood of x1 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)) . x1) = (L . (x - x1)) + (R . (x - x1)) ) ) by A2, Def1;
consider N being Neighbourhood of x1 such that
A4: ( 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)) . x1) = (L . (x - x1)) + (R . (x - x1)) ) by A3;
consider L being LinearFunc, R being RestFunc such that
A5: for x being Real st x in N holds
((SVF1 (1,(pdiff1 (f,1)),u)) . x) - ((SVF1 (1,(pdiff1 (f,1)),u)) . x1) = (L . (x - x1)) + (R . (x - x1)) by A4;
A6: ( x0 = x1 & y0 = y1 & z0 = z1 ) by A1, A3, FINSEQ_1:78;
A7: hpartdiff11 (f,u) = L . 1 by A2, A3, A4, A5, Def10;
SVF1 (1,(pdiff1 (f,1)),u) is_differentiable_in x0 by A4, A6, FDIFF_1:def_4;
hence hpartdiff11 (f,u) = diff ((SVF1 (1,(pdiff1 (f,1)),u)),x0) by A4, A5, A6, A7, FDIFF_1:def_5; ::_thesis: verum
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
let x0, y0, z0 be Real; ::_thesis: 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)
let u be Element of REAL 3; ::_thesis: 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)
let f be PartFunc of (REAL 3),REAL; ::_thesis: ( u = <*x0,y0,z0*> & f is_hpartial_differentiable`12_in u implies hpartdiff12 (f,u) = diff ((SVF1 (2,(pdiff1 (f,1)),u)),y0) )
set r = hpartdiff12 (f,u);
assume that
A1: u = <*x0,y0,z0*> and
A2: f is_hpartial_differentiable`12_in u ; ::_thesis: hpartdiff12 (f,u) = diff ((SVF1 (2,(pdiff1 (f,1)),u)),y0)
consider x1, y1, z1 being Real such that
A3: ( u = <*x1,y1,z1*> & ex N being Neighbourhood of y1 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)) . y1) = (L . (y - y1)) + (R . (y - y1)) ) ) by A2, Def2;
consider N being Neighbourhood of y1 such that
A4: ( 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)) . y1) = (L . (y - y1)) + (R . (y - y1)) ) by A3;
consider L being LinearFunc, R being RestFunc such that
A5: for y being Real st y in N holds
((SVF1 (2,(pdiff1 (f,1)),u)) . y) - ((SVF1 (2,(pdiff1 (f,1)),u)) . y1) = (L . (y - y1)) + (R . (y - y1)) by A4;
A6: ( x0 = x1 & y0 = y1 & z0 = z1 ) by A1, A3, FINSEQ_1:78;
A7: hpartdiff12 (f,u) = L . 1 by A2, A3, A4, A5, Def11;
SVF1 (2,(pdiff1 (f,1)),u) is_differentiable_in y0 by A4, A6, FDIFF_1:def_4;
hence hpartdiff12 (f,u) = diff ((SVF1 (2,(pdiff1 (f,1)),u)),y0) by A4, A5, A6, A7, FDIFF_1:def_5; ::_thesis: verum
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
let x0, y0, z0 be Real; ::_thesis: 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)
let u be Element of REAL 3; ::_thesis: 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)
let f be PartFunc of (REAL 3),REAL; ::_thesis: ( u = <*x0,y0,z0*> & f is_hpartial_differentiable`13_in u implies hpartdiff13 (f,u) = diff ((SVF1 (3,(pdiff1 (f,1)),u)),z0) )
set r = hpartdiff13 (f,u);
assume that
A1: u = <*x0,y0,z0*> and
A2: f is_hpartial_differentiable`13_in u ; ::_thesis: hpartdiff13 (f,u) = diff ((SVF1 (3,(pdiff1 (f,1)),u)),z0)
consider x1, y1, z1 being Real such that
A3: ( u = <*x1,y1,z1*> & ex N being Neighbourhood of z1 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)) . z1) = (L . (z - z1)) + (R . (z - z1)) ) ) by A2, Def3;
consider N being Neighbourhood of z1 such that
A4: ( 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)) . z1) = (L . (z - z1)) + (R . (z - z1)) ) by A3;
consider L being LinearFunc, R being RestFunc such that
A5: for z being Real st z in N holds
((SVF1 (3,(pdiff1 (f,1)),u)) . z) - ((SVF1 (3,(pdiff1 (f,1)),u)) . z1) = (L . (z - z1)) + (R . (z - z1)) by A4;
A6: ( x0 = x1 & y0 = y1 & z0 = z1 ) by A1, A3, FINSEQ_1:78;
A7: hpartdiff13 (f,u) = L . 1 by A2, A3, A4, A5, Def12;
SVF1 (3,(pdiff1 (f,1)),u) is_differentiable_in z0 by A4, A6, FDIFF_1:def_4;
hence hpartdiff13 (f,u) = diff ((SVF1 (3,(pdiff1 (f,1)),u)),z0) by A4, A5, A6, A7, FDIFF_1:def_5; ::_thesis: verum
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
let x0, y0, z0 be Real; ::_thesis: 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)
let u be Element of REAL 3; ::_thesis: 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)
let f be PartFunc of (REAL 3),REAL; ::_thesis: ( u = <*x0,y0,z0*> & f is_hpartial_differentiable`21_in u implies hpartdiff21 (f,u) = diff ((SVF1 (1,(pdiff1 (f,2)),u)),x0) )
set r = hpartdiff21 (f,u);
assume that
A1: u = <*x0,y0,z0*> and
A2: f is_hpartial_differentiable`21_in u ; ::_thesis: hpartdiff21 (f,u) = diff ((SVF1 (1,(pdiff1 (f,2)),u)),x0)
consider x1, y1, z1 being Real such that
A3: ( u = <*x1,y1,z1*> & ex N being Neighbourhood of x1 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)) . x1) = (L . (x - x1)) + (R . (x - x1)) ) ) by A2, Def4;
consider N being Neighbourhood of x1 such that
A4: ( 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)) . x1) = (L . (x - x1)) + (R . (x - x1)) ) by A3;
consider L being LinearFunc, R being RestFunc such that
A5: for x being Real st x in N holds
((SVF1 (1,(pdiff1 (f,2)),u)) . x) - ((SVF1 (1,(pdiff1 (f,2)),u)) . x1) = (L . (x - x1)) + (R . (x - x1)) by A4;
A6: ( x0 = x1 & y0 = y1 & z0 = z1 ) by A1, A3, FINSEQ_1:78;
A7: hpartdiff21 (f,u) = L . 1 by A2, A3, A4, A5, Def13;
SVF1 (1,(pdiff1 (f,2)),u) is_differentiable_in x0 by A4, A6, FDIFF_1:def_4;
hence hpartdiff21 (f,u) = diff ((SVF1 (1,(pdiff1 (f,2)),u)),x0) by A4, A5, A6, A7, FDIFF_1:def_5; ::_thesis: verum
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
let x0, y0, z0 be Real; ::_thesis: 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)
let u be Element of REAL 3; ::_thesis: 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)
let f be PartFunc of (REAL 3),REAL; ::_thesis: ( u = <*x0,y0,z0*> & f is_hpartial_differentiable`22_in u implies hpartdiff22 (f,u) = diff ((SVF1 (2,(pdiff1 (f,2)),u)),y0) )
set r = hpartdiff22 (f,u);
assume that
A1: u = <*x0,y0,z0*> and
A2: f is_hpartial_differentiable`22_in u ; ::_thesis: hpartdiff22 (f,u) = diff ((SVF1 (2,(pdiff1 (f,2)),u)),y0)
consider x1, y1, z1 being Real such that
A3: ( u = <*x1,y1,z1*> & ex N being Neighbourhood of y1 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)) . y1) = (L . (y - y1)) + (R . (y - y1)) ) ) by A2, Def5;
consider N being Neighbourhood of y1 such that
A4: ( 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)) . y1) = (L . (y - y1)) + (R . (y - y1)) ) by A3;
consider L being LinearFunc, R being RestFunc such that
A5: for y being Real st y in N holds
((SVF1 (2,(pdiff1 (f,2)),u)) . y) - ((SVF1 (2,(pdiff1 (f,2)),u)) . y1) = (L . (y - y1)) + (R . (y - y1)) by A4;
A6: ( x0 = x1 & y0 = y1 & z0 = z1 ) by A1, A3, FINSEQ_1:78;
A7: hpartdiff22 (f,u) = L . 1 by A2, A3, A4, A5, Def14;
SVF1 (2,(pdiff1 (f,2)),u) is_differentiable_in y0 by A4, A6, FDIFF_1:def_4;
hence hpartdiff22 (f,u) = diff ((SVF1 (2,(pdiff1 (f,2)),u)),y0) by A4, A5, A6, A7, FDIFF_1:def_5; ::_thesis: verum
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
let x0, y0, z0 be Real; ::_thesis: 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)
let u be Element of REAL 3; ::_thesis: 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)
let f be PartFunc of (REAL 3),REAL; ::_thesis: ( u = <*x0,y0,z0*> & f is_hpartial_differentiable`23_in u implies hpartdiff23 (f,u) = diff ((SVF1 (3,(pdiff1 (f,2)),u)),z0) )
set r = hpartdiff23 (f,u);
assume that
A1: u = <*x0,y0,z0*> and
A2: f is_hpartial_differentiable`23_in u ; ::_thesis: hpartdiff23 (f,u) = diff ((SVF1 (3,(pdiff1 (f,2)),u)),z0)
consider x1, y1, z1 being Real such that
A3: ( u = <*x1,y1,z1*> & ex N being Neighbourhood of z1 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)) . z1) = (L . (z - z1)) + (R . (z - z1)) ) ) by A2, Def6;
consider N being Neighbourhood of z1 such that
A4: ( 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)) . z1) = (L . (z - z1)) + (R . (z - z1)) ) by A3;
consider L being LinearFunc, R being RestFunc such that
A5: for z being Real st z in N holds
((SVF1 (3,(pdiff1 (f,2)),u)) . z) - ((SVF1 (3,(pdiff1 (f,2)),u)) . z1) = (L . (z - z1)) + (R . (z - z1)) by A4;
A6: ( x0 = x1 & y0 = y1 & z0 = z1 ) by A1, A3, FINSEQ_1:78;
A7: hpartdiff23 (f,u) = L . 1 by A2, A3, A4, A5, Def15;
SVF1 (3,(pdiff1 (f,2)),u) is_differentiable_in z0 by A4, A6, FDIFF_1:def_4;
hence hpartdiff23 (f,u) = diff ((SVF1 (3,(pdiff1 (f,2)),u)),z0) by A4, A5, A6, A7, FDIFF_1:def_5; ::_thesis: verum
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
let x0, y0, z0 be Real; ::_thesis: 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)
let u be Element of REAL 3; ::_thesis: 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)
let f be PartFunc of (REAL 3),REAL; ::_thesis: ( u = <*x0,y0,z0*> & f is_hpartial_differentiable`31_in u implies hpartdiff31 (f,u) = diff ((SVF1 (1,(pdiff1 (f,3)),u)),x0) )
set r = hpartdiff31 (f,u);
assume that
A1: u = <*x0,y0,z0*> and
A2: f is_hpartial_differentiable`31_in u ; ::_thesis: hpartdiff31 (f,u) = diff ((SVF1 (1,(pdiff1 (f,3)),u)),x0)
consider x1, y1, z1 being Real such that
A3: ( u = <*x1,y1,z1*> & ex N being Neighbourhood of x1 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)) . x1) = (L . (x - x1)) + (R . (x - x1)) ) ) by A2, Def7;
consider N being Neighbourhood of x1 such that
A4: ( 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)) . x1) = (L . (x - x1)) + (R . (x - x1)) ) by A3;
consider L being LinearFunc, R being RestFunc such that
A5: for x being Real st x in N holds
((SVF1 (1,(pdiff1 (f,3)),u)) . x) - ((SVF1 (1,(pdiff1 (f,3)),u)) . x1) = (L . (x - x1)) + (R . (x - x1)) by A4;
A6: ( x0 = x1 & y0 = y1 & z0 = z1 ) by A1, A3, FINSEQ_1:78;
A7: hpartdiff31 (f,u) = L . 1 by A2, A3, A4, A5, Def16;
SVF1 (1,(pdiff1 (f,3)),u) is_differentiable_in x0 by A4, A6, FDIFF_1:def_4;
hence hpartdiff31 (f,u) = diff ((SVF1 (1,(pdiff1 (f,3)),u)),x0) by A4, A5, A6, A7, FDIFF_1:def_5; ::_thesis: verum
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
let x0, y0, z0 be Real; ::_thesis: 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)
let u be Element of REAL 3; ::_thesis: 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)
let f be PartFunc of (REAL 3),REAL; ::_thesis: ( u = <*x0,y0,z0*> & f is_hpartial_differentiable`32_in u implies hpartdiff32 (f,u) = diff ((SVF1 (2,(pdiff1 (f,3)),u)),y0) )
set r = hpartdiff32 (f,u);
assume that
A1: u = <*x0,y0,z0*> and
A2: f is_hpartial_differentiable`32_in u ; ::_thesis: hpartdiff32 (f,u) = diff ((SVF1 (2,(pdiff1 (f,3)),u)),y0)
consider x1, y1, z1 being Real such that
A3: ( u = <*x1,y1,z1*> & ex N being Neighbourhood of y1 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)) . y1) = (L . (y - y1)) + (R . (y - y1)) ) ) by A2, Def8;
consider N being Neighbourhood of y1 such that
A4: ( 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)) . y1) = (L . (y - y1)) + (R . (y - y1)) ) by A3;
consider L being LinearFunc, R being RestFunc such that
A5: for y being Real st y in N holds
((SVF1 (2,(pdiff1 (f,3)),u)) . y) - ((SVF1 (2,(pdiff1 (f,3)),u)) . y1) = (L . (y - y1)) + (R . (y - y1)) by A4;
A6: ( x0 = x1 & y0 = y1 & z0 = z1 ) by A1, A3, FINSEQ_1:78;
A7: hpartdiff32 (f,u) = L . 1 by A2, A3, A4, A5, Def17;
SVF1 (2,(pdiff1 (f,3)),u) is_differentiable_in y0 by A4, A6, FDIFF_1:def_4;
hence hpartdiff32 (f,u) = diff ((SVF1 (2,(pdiff1 (f,3)),u)),y0) by A4, A5, A6, A7, FDIFF_1:def_5; ::_thesis: verum
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
let x0, y0, z0 be Real; ::_thesis: 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)
let u be Element of REAL 3; ::_thesis: 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)
let f be PartFunc of (REAL 3),REAL; ::_thesis: ( u = <*x0,y0,z0*> & f is_hpartial_differentiable`33_in u implies hpartdiff33 (f,u) = diff ((SVF1 (3,(pdiff1 (f,3)),u)),z0) )
set r = hpartdiff33 (f,u);
assume that
A1: u = <*x0,y0,z0*> and
A2: f is_hpartial_differentiable`33_in u ; ::_thesis: hpartdiff33 (f,u) = diff ((SVF1 (3,(pdiff1 (f,3)),u)),z0)
consider x1, y1, z1 being Real such that
A3: ( u = <*x1,y1,z1*> & ex N being Neighbourhood of z1 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)) . z1) = (L . (z - z1)) + (R . (z - z1)) ) ) by A2, Def9;
consider N being Neighbourhood of z1 such that
A4: ( 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)) . z1) = (L . (z - z1)) + (R . (z - z1)) ) by A3;
consider L being LinearFunc, R being RestFunc such that
A5: for z being Real st z in N holds
((SVF1 (3,(pdiff1 (f,3)),u)) . z) - ((SVF1 (3,(pdiff1 (f,3)),u)) . z1) = (L . (z - z1)) + (R . (z - z1)) by A4;
A6: ( x0 = x1 & y0 = y1 & z0 = z1 ) by A1, A3, FINSEQ_1:78;
A7: hpartdiff33 (f,u) = L . 1 by A2, A3, A4, A5, Def18;
SVF1 (3,(pdiff1 (f,3)),u) is_differentiable_in z0 by A4, A6, FDIFF_1:def_4;
hence hpartdiff33 (f,u) = diff ((SVF1 (3,(pdiff1 (f,3)),u)),z0) by A4, A5, A6, A7, FDIFF_1:def_5; ::_thesis: verum
end;
definition
let f be PartFunc of (REAL 3),REAL;
let D be set ;
predf is_hpartial_differentiable`11_on D means :Def19: :: 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 ) );
predf is_hpartial_differentiable`12_on D means :Def20: :: 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 ) );
predf is_hpartial_differentiable`13_on D means :Def21: :: 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 ) );
predf is_hpartial_differentiable`21_on D means :Def22: :: 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 ) );
predf is_hpartial_differentiable`22_on D means :Def23: :: 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 ) );
predf is_hpartial_differentiable`23_on D means :Def24: :: 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 ) );
predf is_hpartial_differentiable`31_on D means :Def25: :: 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 ) );
predf is_hpartial_differentiable`32_on D means :Def26: :: 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 ) );
predf is_hpartial_differentiable`33_on D means :Def27: :: 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 Def19 defines is_hpartial_differentiable`11_on PDIFF_5:def_19_:_
for f being PartFunc of (REAL 3),REAL
for D being set holds
( f is_hpartial_differentiable`11_on D iff ( D c= dom f & ( for u being Element of REAL 3 st u in D holds
f | D is_hpartial_differentiable`11_in u ) ) );
:: deftheorem Def20 defines is_hpartial_differentiable`12_on PDIFF_5:def_20_:_
for f being PartFunc of (REAL 3),REAL
for D being set holds
( f is_hpartial_differentiable`12_on D iff ( D c= dom f & ( for u being Element of REAL 3 st u in D holds
f | D is_hpartial_differentiable`12_in u ) ) );
:: deftheorem Def21 defines is_hpartial_differentiable`13_on PDIFF_5:def_21_:_
for f being PartFunc of (REAL 3),REAL
for D being set holds
( f is_hpartial_differentiable`13_on D iff ( D c= dom f & ( for u being Element of REAL 3 st u in D holds
f | D is_hpartial_differentiable`13_in u ) ) );
:: deftheorem Def22 defines is_hpartial_differentiable`21_on PDIFF_5:def_22_:_
for f being PartFunc of (REAL 3),REAL
for D being set holds
( f is_hpartial_differentiable`21_on D iff ( D c= dom f & ( for u being Element of REAL 3 st u in D holds
f | D is_hpartial_differentiable`21_in u ) ) );
:: deftheorem Def23 defines is_hpartial_differentiable`22_on PDIFF_5:def_23_:_
for f being PartFunc of (REAL 3),REAL
for D being set holds
( f is_hpartial_differentiable`22_on D iff ( D c= dom f & ( for u being Element of REAL 3 st u in D holds
f | D is_hpartial_differentiable`22_in u ) ) );
:: deftheorem Def24 defines is_hpartial_differentiable`23_on PDIFF_5:def_24_:_
for f being PartFunc of (REAL 3),REAL
for D being set holds
( f is_hpartial_differentiable`23_on D iff ( D c= dom f & ( for u being Element of REAL 3 st u in D holds
f | D is_hpartial_differentiable`23_in u ) ) );
:: deftheorem Def25 defines is_hpartial_differentiable`31_on PDIFF_5:def_25_:_
for f being PartFunc of (REAL 3),REAL
for D being set holds
( f is_hpartial_differentiable`31_on D iff ( D c= dom f & ( for u being Element of REAL 3 st u in D holds
f | D is_hpartial_differentiable`31_in u ) ) );
:: deftheorem Def26 defines is_hpartial_differentiable`32_on PDIFF_5:def_26_:_
for f being PartFunc of (REAL 3),REAL
for D being set holds
( f is_hpartial_differentiable`32_on D iff ( D c= dom f & ( for u being Element of REAL 3 st u in D holds
f | D is_hpartial_differentiable`32_in u ) ) );
:: deftheorem Def27 defines is_hpartial_differentiable`33_on PDIFF_5:def_27_:_
for f being PartFunc of (REAL 3),REAL
for D being set holds
( f is_hpartial_differentiable`33_on D iff ( D c= dom f & ( for u being Element of REAL 3 st u in D holds
f | D is_hpartial_differentiable`33_in u ) ) );
definition
let f be PartFunc of (REAL 3),REAL;
let D be set ;
assume A1: f is_hpartial_differentiable`11_on D ;
funcf `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
defpred S1[ Element of REAL 3] means $1 in D;
deffunc H1( Element of REAL 3) -> Real = hpartdiff11 (f,$1);
consider F being PartFunc of (REAL 3),REAL such that
A2: ( ( for u being Element of REAL 3 holds
( u in dom F iff S1[u] ) ) & ( for u being Element of REAL 3 st u in dom F holds
F . u = H1(u) ) ) from SEQ_1:sch_3();
take F ; ::_thesis: ( dom F = D & ( for u being Element of REAL 3 st u in D holds
F . u = hpartdiff11 (f,u) ) )
for y being set st y in dom F holds
y in D by A2;
then A3: dom F c= D by TARSKI:def_3;
now__::_thesis:_for_y_being_set_st_y_in_D_holds_
y_in_dom_F
let y be set ; ::_thesis: ( y in D implies y in dom F )
assume A4: y in D ; ::_thesis: y in dom F
D c= dom f by A1, Def19;
then D is Subset of (REAL 3) by XBOOLE_1:1;
hence y in dom F by A2, A4; ::_thesis: verum
end;
then D c= dom F by TARSKI:def_3;
hence dom F = D by A3, XBOOLE_0:def_10; ::_thesis: for u being Element of REAL 3 st u in D holds
F . u = hpartdiff11 (f,u)
hereby ::_thesis: verum
let u be Element of REAL 3; ::_thesis: ( u in D implies F . u = hpartdiff11 (f,u) )
assume u in D ; ::_thesis: F . u = hpartdiff11 (f,u)
then u in dom F by A2;
hence F . u = hpartdiff11 (f,u) by A2; ::_thesis: verum
end;
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
let F, G be PartFunc of (REAL 3),REAL; ::_thesis: ( dom F = D & ( for u being Element of REAL 3 st u in D holds
F . u = hpartdiff11 (f,u) ) & dom G = D & ( for u being Element of REAL 3 st u in D holds
G . u = hpartdiff11 (f,u) ) implies F = G )
assume that
A5: ( dom F = D & ( for u being Element of REAL 3 st u in D holds
F . u = hpartdiff11 (f,u) ) ) and
A6: ( dom G = D & ( for u being Element of REAL 3 st u in D holds
G . u = hpartdiff11 (f,u) ) ) ; ::_thesis: F = G
now__::_thesis:_for_u_being_Element_of_REAL_3_st_u_in_dom_F_holds_
F_._u_=_G_._u
let u be Element of REAL 3; ::_thesis: ( u in dom F implies F . u = G . u )
assume A7: u in dom F ; ::_thesis: F . u = G . u
then F . u = hpartdiff11 (f,u) by A5;
hence F . u = G . u by A5, A6, A7; ::_thesis: verum
end;
hence F = G by A5, A6, PARTFUN1:5; ::_thesis: verum
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_differentiable`11_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_differentiable`12_on D ;
funcf `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
defpred S1[ Element of REAL 3] means $1 in D;
deffunc H1( Element of REAL 3) -> Real = hpartdiff12 (f,$1);
consider F being PartFunc of (REAL 3),REAL such that
A2: ( ( for u being Element of REAL 3 holds
( u in dom F iff S1[u] ) ) & ( for u being Element of REAL 3 st u in dom F holds
F . u = H1(u) ) ) from SEQ_1:sch_3();
take F ; ::_thesis: ( dom F = D & ( for u being Element of REAL 3 st u in D holds
F . u = hpartdiff12 (f,u) ) )
for y being set st y in dom F holds
y in D by A2;
then A3: dom F c= D by TARSKI:def_3;
now__::_thesis:_for_y_being_set_st_y_in_D_holds_
y_in_dom_F
let y be set ; ::_thesis: ( y in D implies y in dom F )
assume A4: y in D ; ::_thesis: y in dom F
D c= dom f by A1, Def20;
then D is Subset of (REAL 3) by XBOOLE_1:1;
hence y in dom F by A2, A4; ::_thesis: verum
end;
then D c= dom F by TARSKI:def_3;
hence dom F = D by A3, XBOOLE_0:def_10; ::_thesis: for u being Element of REAL 3 st u in D holds
F . u = hpartdiff12 (f,u)
hereby ::_thesis: verum
let u be Element of REAL 3; ::_thesis: ( u in D implies F . u = hpartdiff12 (f,u) )
assume u in D ; ::_thesis: F . u = hpartdiff12 (f,u)
then u in dom F by A2;
hence F . u = hpartdiff12 (f,u) by A2; ::_thesis: verum
end;
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
let F, G be PartFunc of (REAL 3),REAL; ::_thesis: ( dom F = D & ( for u being Element of REAL 3 st u in D holds
F . u = hpartdiff12 (f,u) ) & dom G = D & ( for u being Element of REAL 3 st u in D holds
G . u = hpartdiff12 (f,u) ) implies F = G )
assume that
A5: ( dom F = D & ( for u being Element of REAL 3 st u in D holds
F . u = hpartdiff12 (f,u) ) ) and
A6: ( dom G = D & ( for u being Element of REAL 3 st u in D holds
G . u = hpartdiff12 (f,u) ) ) ; ::_thesis: F = G
now__::_thesis:_for_u_being_Element_of_REAL_3_st_u_in_dom_F_holds_
F_._u_=_G_._u
let u be Element of REAL 3; ::_thesis: ( u in dom F implies F . u = G . u )
assume A7: u in dom F ; ::_thesis: F . u = G . u
then F . u = hpartdiff12 (f,u) by A5;
hence F . u = G . u by A5, A6, A7; ::_thesis: verum
end;
hence F = G by A5, A6, PARTFUN1:5; ::_thesis: verum
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_differentiable`12_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_differentiable`13_on D ;
funcf `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
defpred S1[ Element of REAL 3] means $1 in D;
deffunc H1( Element of REAL 3) -> Real = hpartdiff13 (f,$1);
consider F being PartFunc of (REAL 3),REAL such that
A2: ( ( for u being Element of REAL 3 holds
( u in dom F iff S1[u] ) ) & ( for u being Element of REAL 3 st u in dom F holds
F . u = H1(u) ) ) from SEQ_1:sch_3();
take F ; ::_thesis: ( dom F = D & ( for u being Element of REAL 3 st u in D holds
F . u = hpartdiff13 (f,u) ) )
for y being set st y in dom F holds
y in D by A2;
then A3: dom F c= D by TARSKI:def_3;
now__::_thesis:_for_y_being_set_st_y_in_D_holds_
y_in_dom_F
let y be set ; ::_thesis: ( y in D implies y in dom F )
assume A4: y in D ; ::_thesis: y in dom F
D c= dom f by A1, Def21;
then D is Subset of (REAL 3) by XBOOLE_1:1;
hence y in dom F by A2, A4; ::_thesis: verum
end;
then D c= dom F by TARSKI:def_3;
hence dom F = D by A3, XBOOLE_0:def_10; ::_thesis: for u being Element of REAL 3 st u in D holds
F . u = hpartdiff13 (f,u)
hereby ::_thesis: verum
let u be Element of REAL 3; ::_thesis: ( u in D implies F . u = hpartdiff13 (f,u) )
assume u in D ; ::_thesis: F . u = hpartdiff13 (f,u)
then u in dom F by A2;
hence F . u = hpartdiff13 (f,u) by A2; ::_thesis: verum
end;
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
let F, G be PartFunc of (REAL 3),REAL; ::_thesis: ( dom F = D & ( for u being Element of REAL 3 st u in D holds
F . u = hpartdiff13 (f,u) ) & dom G = D & ( for u being Element of REAL 3 st u in D holds
G . u = hpartdiff13 (f,u) ) implies F = G )
assume that
A5: ( dom F = D & ( for u being Element of REAL 3 st u in D holds
F . u = hpartdiff13 (f,u) ) ) and
A6: ( dom G = D & ( for u being Element of REAL 3 st u in D holds
G . u = hpartdiff13 (f,u) ) ) ; ::_thesis: F = G
now__::_thesis:_for_u_being_Element_of_REAL_3_st_u_in_dom_F_holds_
F_._u_=_G_._u
let u be Element of REAL 3; ::_thesis: ( u in dom F implies F . u = G . u )
assume A7: u in dom F ; ::_thesis: F . u = G . u
then F . u = hpartdiff13 (f,u) by A5;
hence F . u = G . u by A5, A6, A7; ::_thesis: verum
end;
hence F = G by A5, A6, PARTFUN1:5; ::_thesis: verum
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_differentiable`13_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_differentiable`21_on D ;
funcf `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
defpred S1[ Element of REAL 3] means $1 in D;
deffunc H1( Element of REAL 3) -> Real = hpartdiff21 (f,$1);
consider F being PartFunc of (REAL 3),REAL such that
A2: ( ( for u being Element of REAL 3 holds
( u in dom F iff S1[u] ) ) & ( for u being Element of REAL 3 st u in dom F holds
F . u = H1(u) ) ) from SEQ_1:sch_3();
take F ; ::_thesis: ( dom F = D & ( for u being Element of REAL 3 st u in D holds
F . u = hpartdiff21 (f,u) ) )
for y being set st y in dom F holds
y in D by A2;
then A3: dom F c= D by TARSKI:def_3;
now__::_thesis:_for_y_being_set_st_y_in_D_holds_
y_in_dom_F
let y be set ; ::_thesis: ( y in D implies y in dom F )
assume A4: y in D ; ::_thesis: y in dom F
D c= dom f by A1, Def22;
then D is Subset of (REAL 3) by XBOOLE_1:1;
hence y in dom F by A2, A4; ::_thesis: verum
end;
then D c= dom F by TARSKI:def_3;
hence dom F = D by A3, XBOOLE_0:def_10; ::_thesis: for u being Element of REAL 3 st u in D holds
F . u = hpartdiff21 (f,u)
hereby ::_thesis: verum
let u be Element of REAL 3; ::_thesis: ( u in D implies F . u = hpartdiff21 (f,u) )
assume u in D ; ::_thesis: F . u = hpartdiff21 (f,u)
then u in dom F by A2;
hence F . u = hpartdiff21 (f,u) by A2; ::_thesis: verum
end;
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
let F, G be PartFunc of (REAL 3),REAL; ::_thesis: ( dom F = D & ( for u being Element of REAL 3 st u in D holds
F . u = hpartdiff21 (f,u) ) & dom G = D & ( for u being Element of REAL 3 st u in D holds
G . u = hpartdiff21 (f,u) ) implies F = G )
assume that
A5: ( dom F = D & ( for u being Element of REAL 3 st u in D holds
F . u = hpartdiff21 (f,u) ) ) and
A6: ( dom G = D & ( for u being Element of REAL 3 st u in D holds
G . u = hpartdiff21 (f,u) ) ) ; ::_thesis: F = G
now__::_thesis:_for_u_being_Element_of_REAL_3_st_u_in_dom_F_holds_
F_._u_=_G_._u
let u be Element of REAL 3; ::_thesis: ( u in dom F implies F . u = G . u )
assume A7: u in dom F ; ::_thesis: F . u = G . u
then F . u = hpartdiff21 (f,u) by A5;
hence F . u = G . u by A5, A6, A7; ::_thesis: verum
end;
hence F = G by A5, A6, PARTFUN1:5; ::_thesis: verum
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_differentiable`21_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_differentiable`22_on D ;
funcf `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
defpred S1[ Element of REAL 3] means $1 in D;
deffunc H1( Element of REAL 3) -> Real = hpartdiff22 (f,$1);
consider F being PartFunc of (REAL 3),REAL such that
A2: ( ( for u being Element of REAL 3 holds
( u in dom F iff S1[u] ) ) & ( for u being Element of REAL 3 st u in dom F holds
F . u = H1(u) ) ) from SEQ_1:sch_3();
take F ; ::_thesis: ( dom F = D & ( for u being Element of REAL 3 st u in D holds
F . u = hpartdiff22 (f,u) ) )
for y being set st y in dom F holds
y in D by A2;
then A3: dom F c= D by TARSKI:def_3;
now__::_thesis:_for_y_being_set_st_y_in_D_holds_
y_in_dom_F
let y be set ; ::_thesis: ( y in D implies y in dom F )
assume A4: y in D ; ::_thesis: y in dom F
D c= dom f by A1, Def23;
then D is Subset of (REAL 3) by XBOOLE_1:1;
hence y in dom F by A2, A4; ::_thesis: verum
end;
then D c= dom F by TARSKI:def_3;
hence dom F = D by A3, XBOOLE_0:def_10; ::_thesis: for u being Element of REAL 3 st u in D holds
F . u = hpartdiff22 (f,u)
hereby ::_thesis: verum
let u be Element of REAL 3; ::_thesis: ( u in D implies F . u = hpartdiff22 (f,u) )
assume u in D ; ::_thesis: F . u = hpartdiff22 (f,u)
then u in dom F by A2;
hence F . u = hpartdiff22 (f,u) by A2; ::_thesis: verum
end;
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
let F, G be PartFunc of (REAL 3),REAL; ::_thesis: ( dom F = D & ( for u being Element of REAL 3 st u in D holds
F . u = hpartdiff22 (f,u) ) & dom G = D & ( for u being Element of REAL 3 st u in D holds
G . u = hpartdiff22 (f,u) ) implies F = G )
assume that
A5: ( dom F = D & ( for u being Element of REAL 3 st u in D holds
F . u = hpartdiff22 (f,u) ) ) and
A6: ( dom G = D & ( for u being Element of REAL 3 st u in D holds
G . u = hpartdiff22 (f,u) ) ) ; ::_thesis: F = G
now__::_thesis:_for_u_being_Element_of_REAL_3_st_u_in_dom_F_holds_
F_._u_=_G_._u
let u be Element of REAL 3; ::_thesis: ( u in dom F implies F . u = G . u )
assume A7: u in dom F ; ::_thesis: F . u = G . u
then F . u = hpartdiff22 (f,u) by A5;
hence F . u = G . u by A5, A6, A7; ::_thesis: verum
end;
hence F = G by A5, A6, PARTFUN1:5; ::_thesis: verum
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_differentiable`22_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_differentiable`23_on D ;
funcf `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
defpred S1[ Element of REAL 3] means $1 in D;
deffunc H1( Element of REAL 3) -> Real = hpartdiff23 (f,$1);
consider F being PartFunc of (REAL 3),REAL such that
A2: ( ( for u being Element of REAL 3 holds
( u in dom F iff S1[u] ) ) & ( for u being Element of REAL 3 st u in dom F holds
F . u = H1(u) ) ) from SEQ_1:sch_3();
take F ; ::_thesis: ( dom F = D & ( for u being Element of REAL 3 st u in D holds
F . u = hpartdiff23 (f,u) ) )
for y being set st y in dom F holds
y in D by A2;
then A3: dom F c= D by TARSKI:def_3;
now__::_thesis:_for_y_being_set_st_y_in_D_holds_
y_in_dom_F
let y be set ; ::_thesis: ( y in D implies y in dom F )
assume A4: y in D ; ::_thesis: y in dom F
D c= dom f by A1, Def24;
then D is Subset of (REAL 3) by XBOOLE_1:1;
hence y in dom F by A2, A4; ::_thesis: verum
end;
then D c= dom F by TARSKI:def_3;
hence dom F = D by A3, XBOOLE_0:def_10; ::_thesis: for u being Element of REAL 3 st u in D holds
F . u = hpartdiff23 (f,u)
hereby ::_thesis: verum
let u be Element of REAL 3; ::_thesis: ( u in D implies F . u = hpartdiff23 (f,u) )
assume u in D ; ::_thesis: F . u = hpartdiff23 (f,u)
then u in dom F by A2;
hence F . u = hpartdiff23 (f,u) by A2; ::_thesis: verum
end;
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
let F, G be PartFunc of (REAL 3),REAL; ::_thesis: ( dom F = D & ( for u being Element of REAL 3 st u in D holds
F . u = hpartdiff23 (f,u) ) & dom G = D & ( for u being Element of REAL 3 st u in D holds
G . u = hpartdiff23 (f,u) ) implies F = G )
assume that
A5: ( dom F = D & ( for u being Element of REAL 3 st u in D holds
F . u = hpartdiff23 (f,u) ) ) and
A6: ( dom G = D & ( for u being Element of REAL 3 st u in D holds
G . u = hpartdiff23 (f,u) ) ) ; ::_thesis: F = G
now__::_thesis:_for_u_being_Element_of_REAL_3_st_u_in_dom_F_holds_
F_._u_=_G_._u
let u be Element of REAL 3; ::_thesis: ( u in dom F implies F . u = G . u )
assume A7: u in dom F ; ::_thesis: F . u = G . u
then F . u = hpartdiff23 (f,u) by A5;
hence F . u = G . u by A5, A6, A7; ::_thesis: verum
end;
hence F = G by A5, A6, PARTFUN1:5; ::_thesis: verum
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_differentiable`23_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_differentiable`31_on D ;
funcf `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
defpred S1[ Element of REAL 3] means $1 in D;
deffunc H1( Element of REAL 3) -> Real = hpartdiff31 (f,$1);
consider F being PartFunc of (REAL 3),REAL such that
A2: ( ( for u being Element of REAL 3 holds
( u in dom F iff S1[u] ) ) & ( for u being Element of REAL 3 st u in dom F holds
F . u = H1(u) ) ) from SEQ_1:sch_3();
take F ; ::_thesis: ( dom F = D & ( for u being Element of REAL 3 st u in D holds
F . u = hpartdiff31 (f,u) ) )
for y being set st y in dom F holds
y in D by A2;
then A3: dom F c= D by TARSKI:def_3;
now__::_thesis:_for_y_being_set_st_y_in_D_holds_
y_in_dom_F
let y be set ; ::_thesis: ( y in D implies y in dom F )
assume A4: y in D ; ::_thesis: y in dom F
D c= dom f by A1, Def25;
then D is Subset of (REAL 3) by XBOOLE_1:1;
hence y in dom F by A2, A4; ::_thesis: verum
end;
then D c= dom F by TARSKI:def_3;
hence dom F = D by A3, XBOOLE_0:def_10; ::_thesis: for u being Element of REAL 3 st u in D holds
F . u = hpartdiff31 (f,u)
hereby ::_thesis: verum
let u be Element of REAL 3; ::_thesis: ( u in D implies F . u = hpartdiff31 (f,u) )
assume u in D ; ::_thesis: F . u = hpartdiff31 (f,u)
then u in dom F by A2;
hence F . u = hpartdiff31 (f,u) by A2; ::_thesis: verum
end;
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
let F, G be PartFunc of (REAL 3),REAL; ::_thesis: ( dom F = D & ( for u being Element of REAL 3 st u in D holds
F . u = hpartdiff31 (f,u) ) & dom G = D & ( for u being Element of REAL 3 st u in D holds
G . u = hpartdiff31 (f,u) ) implies F = G )
assume that
A5: ( dom F = D & ( for u being Element of REAL 3 st u in D holds
F . u = hpartdiff31 (f,u) ) ) and
A6: ( dom G = D & ( for u being Element of REAL 3 st u in D holds
G . u = hpartdiff31 (f,u) ) ) ; ::_thesis: F = G
now__::_thesis:_for_u_being_Element_of_REAL_3_st_u_in_dom_F_holds_
F_._u_=_G_._u
let u be Element of REAL 3; ::_thesis: ( u in dom F implies F . u = G . u )
assume A7: u in dom F ; ::_thesis: F . u = G . u
then F . u = hpartdiff31 (f,u) by A5;
hence F . u = G . u by A5, A6, A7; ::_thesis: verum
end;
hence F = G by A5, A6, PARTFUN1:5; ::_thesis: verum
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_differentiable`31_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_differentiable`32_on D ;
funcf `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
defpred S1[ Element of REAL 3] means $1 in D;
deffunc H1( Element of REAL 3) -> Real = hpartdiff32 (f,$1);
consider F being PartFunc of (REAL 3),REAL such that
A2: ( ( for u being Element of REAL 3 holds
( u in dom F iff S1[u] ) ) & ( for u being Element of REAL 3 st u in dom F holds
F . u = H1(u) ) ) from SEQ_1:sch_3();
take F ; ::_thesis: ( dom F = D & ( for u being Element of REAL 3 st u in D holds
F . u = hpartdiff32 (f,u) ) )
for y being set st y in dom F holds
y in D by A2;
then A3: dom F c= D by TARSKI:def_3;
now__::_thesis:_for_y_being_set_st_y_in_D_holds_
y_in_dom_F
let y be set ; ::_thesis: ( y in D implies y in dom F )
assume A4: y in D ; ::_thesis: y in dom F
D c= dom f by A1, Def26;
then D is Subset of (REAL 3) by XBOOLE_1:1;
hence y in dom F by A2, A4; ::_thesis: verum
end;
then D c= dom F by TARSKI:def_3;
hence dom F = D by A3, XBOOLE_0:def_10; ::_thesis: for u being Element of REAL 3 st u in D holds
F . u = hpartdiff32 (f,u)
hereby ::_thesis: verum
let u be Element of REAL 3; ::_thesis: ( u in D implies F . u = hpartdiff32 (f,u) )
assume u in D ; ::_thesis: F . u = hpartdiff32 (f,u)
then u in dom F by A2;
hence F . u = hpartdiff32 (f,u) by A2; ::_thesis: verum
end;
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
let F, G be PartFunc of (REAL 3),REAL; ::_thesis: ( dom F = D & ( for u being Element of REAL 3 st u in D holds
F . u = hpartdiff32 (f,u) ) & dom G = D & ( for u being Element of REAL 3 st u in D holds
G . u = hpartdiff32 (f,u) ) implies F = G )
assume that
A5: ( dom F = D & ( for u being Element of REAL 3 st u in D holds
F . u = hpartdiff32 (f,u) ) ) and
A6: ( dom G = D & ( for u being Element of REAL 3 st u in D holds
G . u = hpartdiff32 (f,u) ) ) ; ::_thesis: F = G
now__::_thesis:_for_u_being_Element_of_REAL_3_st_u_in_dom_F_holds_
F_._u_=_G_._u
let u be Element of REAL 3; ::_thesis: ( u in dom F implies F . u = G . u )
assume A7: u in dom F ; ::_thesis: F . u = G . u
then F . u = hpartdiff32 (f,u) by A5;
hence F . u = G . u by A5, A6, A7; ::_thesis: verum
end;
hence F = G by A5, A6, PARTFUN1:5; ::_thesis: verum
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_differentiable`32_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_differentiable`33_on D ;
funcf `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
defpred S1[ Element of REAL 3] means $1 in D;
deffunc H1( Element of REAL 3) -> Real = hpartdiff33 (f,$1);
consider F being PartFunc of (REAL 3),REAL such that
A2: ( ( for u being Element of REAL 3 holds
( u in dom F iff S1[u] ) ) & ( for u being Element of REAL 3 st u in dom F holds
F . u = H1(u) ) ) from SEQ_1:sch_3();
take F ; ::_thesis: ( dom F = D & ( for u being Element of REAL 3 st u in D holds
F . u = hpartdiff33 (f,u) ) )
for y being set st y in dom F holds
y in D by A2;
then A3: dom F c= D by TARSKI:def_3;
now__::_thesis:_for_y_being_set_st_y_in_D_holds_
y_in_dom_F
let y be set ; ::_thesis: ( y in D implies y in dom F )
assume A4: y in D ; ::_thesis: y in dom F
D c= dom f by A1, Def27;
then D is Subset of (REAL 3) by XBOOLE_1:1;
hence y in dom F by A2, A4; ::_thesis: verum
end;
then D c= dom F by TARSKI:def_3;
hence dom F = D by A3, XBOOLE_0:def_10; ::_thesis: for u being Element of REAL 3 st u in D holds
F . u = hpartdiff33 (f,u)
hereby ::_thesis: verum
let u be Element of REAL 3; ::_thesis: ( u in D implies F . u = hpartdiff33 (f,u) )
assume u in D ; ::_thesis: F . u = hpartdiff33 (f,u)
then u in dom F by A2;
hence F . u = hpartdiff33 (f,u) by A2; ::_thesis: verum
end;
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
let F, G be PartFunc of (REAL 3),REAL; ::_thesis: ( dom F = D & ( for u being Element of REAL 3 st u in D holds
F . u = hpartdiff33 (f,u) ) & dom G = D & ( for u being Element of REAL 3 st u in D holds
G . u = hpartdiff33 (f,u) ) implies F = G )
assume that
A5: ( dom F = D & ( for u being Element of REAL 3 st u in D holds
F . u = hpartdiff33 (f,u) ) ) and
A6: ( dom G = D & ( for u being Element of REAL 3 st u in D holds
G . u = hpartdiff33 (f,u) ) ) ; ::_thesis: F = G
now__::_thesis:_for_u_being_Element_of_REAL_3_st_u_in_dom_F_holds_
F_._u_=_G_._u
let u be Element of REAL 3; ::_thesis: ( u in dom F implies F . u = G . u )
assume A7: u in dom F ; ::_thesis: F . u = G . u
then F . u = hpartdiff33 (f,u) by A5;
hence F . u = G . u by A5, A6, A7; ::_thesis: verum
end;
hence F = G by A5, A6, PARTFUN1:5; ::_thesis: verum
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_differentiable`33_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) ) ) );
begin
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 )
proof
let u be Element of REAL 3; ::_thesis: 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 )
let f be PartFunc of (REAL 3),REAL; ::_thesis: ( f is_hpartial_differentiable`11_in u iff pdiff1 (f,1) is_partial_differentiable_in u,1 )
thus ( f is_hpartial_differentiable`11_in u implies pdiff1 (f,1) is_partial_differentiable_in u,1 ) ::_thesis: ( pdiff1 (f,1) is_partial_differentiable_in u,1 implies f is_hpartial_differentiable`11_in u )
proof
assume f is_hpartial_differentiable`11_in u ; ::_thesis: pdiff1 (f,1) is_partial_differentiable_in u,1
then consider x0, y0, z0 being Real such that
A1: ( 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)) ) ) by Def1;
thus pdiff1 (f,1) is_partial_differentiable_in u,1 by A1, PDIFF_4:13; ::_thesis: verum
end;
assume pdiff1 (f,1) is_partial_differentiable_in u,1 ; ::_thesis: f is_hpartial_differentiable`11_in u
then consider x0, y0, z0 being Real such that
A2: ( 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)) ) ) by PDIFF_4:13;
thus f is_hpartial_differentiable`11_in u by A2, Def1; ::_thesis: verum
end;
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 )
proof
let u be Element of REAL 3; ::_thesis: 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 )
let f be PartFunc of (REAL 3),REAL; ::_thesis: ( f is_hpartial_differentiable`12_in u iff pdiff1 (f,1) is_partial_differentiable_in u,2 )
thus ( f is_hpartial_differentiable`12_in u implies pdiff1 (f,1) is_partial_differentiable_in u,2 ) ::_thesis: ( pdiff1 (f,1) is_partial_differentiable_in u,2 implies f is_hpartial_differentiable`12_in u )
proof
assume f is_hpartial_differentiable`12_in u ; ::_thesis: pdiff1 (f,1) is_partial_differentiable_in u,2
then consider x0, y0, z0 being Real such that
A1: ( 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)) ) ) by Def2;
thus pdiff1 (f,1) is_partial_differentiable_in u,2 by A1, PDIFF_4:14; ::_thesis: verum
end;
assume pdiff1 (f,1) is_partial_differentiable_in u,2 ; ::_thesis: f is_hpartial_differentiable`12_in u
then consider x0, y0, z0 being Real such that
A2: ( 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)) ) ) by PDIFF_4:14;
thus f is_hpartial_differentiable`12_in u by A2, Def2; ::_thesis: verum
end;
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 )
proof
let u be Element of REAL 3; ::_thesis: 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 )
let f be PartFunc of (REAL 3),REAL; ::_thesis: ( f is_hpartial_differentiable`13_in u iff pdiff1 (f,1) is_partial_differentiable_in u,3 )
thus ( f is_hpartial_differentiable`13_in u implies pdiff1 (f,1) is_partial_differentiable_in u,3 ) ::_thesis: ( pdiff1 (f,1) is_partial_differentiable_in u,3 implies f is_hpartial_differentiable`13_in u )
proof
assume f is_hpartial_differentiable`13_in u ; ::_thesis: pdiff1 (f,1) is_partial_differentiable_in u,3
then consider x0, y0, z0 being Real such that
A1: ( 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)) ) ) by Def3;
thus pdiff1 (f,1) is_partial_differentiable_in u,3 by A1, PDIFF_4:15; ::_thesis: verum
end;
assume pdiff1 (f,1) is_partial_differentiable_in u,3 ; ::_thesis: f is_hpartial_differentiable`13_in u
then consider x0, y0, z0 being Real such that
A2: ( 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)) ) ) by PDIFF_4:15;
thus f is_hpartial_differentiable`13_in u by A2, Def3; ::_thesis: verum
end;
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 )
proof
let u be Element of REAL 3; ::_thesis: 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 )
let f be PartFunc of (REAL 3),REAL; ::_thesis: ( f is_hpartial_differentiable`21_in u iff pdiff1 (f,2) is_partial_differentiable_in u,1 )
thus ( f is_hpartial_differentiable`21_in u implies pdiff1 (f,2) is_partial_differentiable_in u,1 ) ::_thesis: ( pdiff1 (f,2) is_partial_differentiable_in u,1 implies f is_hpartial_differentiable`21_in u )
proof
assume f is_hpartial_differentiable`21_in u ; ::_thesis: pdiff1 (f,2) is_partial_differentiable_in u,1
then consider x0, y0, z0 being Real such that
A1: ( 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)) ) ) by Def4;
thus pdiff1 (f,2) is_partial_differentiable_in u,1 by A1, PDIFF_4:13; ::_thesis: verum
end;
assume pdiff1 (f,2) is_partial_differentiable_in u,1 ; ::_thesis: f is_hpartial_differentiable`21_in u
then consider x0, y0, z0 being Real such that
A2: ( 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)) ) ) by PDIFF_4:13;
thus f is_hpartial_differentiable`21_in u by A2, Def4; ::_thesis: verum
end;
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 )
proof
let u be Element of REAL 3; ::_thesis: 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 )
let f be PartFunc of (REAL 3),REAL; ::_thesis: ( f is_hpartial_differentiable`22_in u iff pdiff1 (f,2) is_partial_differentiable_in u,2 )
thus ( f is_hpartial_differentiable`22_in u implies pdiff1 (f,2) is_partial_differentiable_in u,2 ) ::_thesis: ( pdiff1 (f,2) is_partial_differentiable_in u,2 implies f is_hpartial_differentiable`22_in u )
proof
assume f is_hpartial_differentiable`22_in u ; ::_thesis: pdiff1 (f,2) is_partial_differentiable_in u,2
then consider x0, y0, z0 being Real such that
A1: ( 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)) ) ) by Def5;
thus pdiff1 (f,2) is_partial_differentiable_in u,2 by A1, PDIFF_4:14; ::_thesis: verum
end;
assume pdiff1 (f,2) is_partial_differentiable_in u,2 ; ::_thesis: f is_hpartial_differentiable`22_in u
then consider x0, y0, z0 being Real such that
A2: ( 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)) ) ) by PDIFF_4:14;
thus f is_hpartial_differentiable`22_in u by A2, Def5; ::_thesis: verum
end;
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 )
proof
let u be Element of REAL 3; ::_thesis: 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 )
let f be PartFunc of (REAL 3),REAL; ::_thesis: ( f is_hpartial_differentiable`23_in u iff pdiff1 (f,2) is_partial_differentiable_in u,3 )
thus ( f is_hpartial_differentiable`23_in u implies pdiff1 (f,2) is_partial_differentiable_in u,3 ) ::_thesis: ( pdiff1 (f,2) is_partial_differentiable_in u,3 implies f is_hpartial_differentiable`23_in u )
proof
assume f is_hpartial_differentiable`23_in u ; ::_thesis: pdiff1 (f,2) is_partial_differentiable_in u,3
then consider x0, y0, z0 being Real such that
A1: ( 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)) ) ) by Def6;
thus pdiff1 (f,2) is_partial_differentiable_in u,3 by A1, PDIFF_4:15; ::_thesis: verum
end;
assume pdiff1 (f,2) is_partial_differentiable_in u,3 ; ::_thesis: f is_hpartial_differentiable`23_in u
then consider x0, y0, z0 being Real such that
A2: ( 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)) ) ) by PDIFF_4:15;
thus f is_hpartial_differentiable`23_in u by A2, Def6; ::_thesis: verum
end;
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 )
proof
let u be Element of REAL 3; ::_thesis: 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 )
let f be PartFunc of (REAL 3),REAL; ::_thesis: ( f is_hpartial_differentiable`31_in u iff pdiff1 (f,3) is_partial_differentiable_in u,1 )
thus ( f is_hpartial_differentiable`31_in u implies pdiff1 (f,3) is_partial_differentiable_in u,1 ) ::_thesis: ( pdiff1 (f,3) is_partial_differentiable_in u,1 implies f is_hpartial_differentiable`31_in u )
proof
assume f is_hpartial_differentiable`31_in u ; ::_thesis: pdiff1 (f,3) is_partial_differentiable_in u,1
then consider x0, y0, z0 being Real such that
A1: ( 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)) ) ) by Def7;
thus pdiff1 (f,3) is_partial_differentiable_in u,1 by A1, PDIFF_4:13; ::_thesis: verum
end;
assume pdiff1 (f,3) is_partial_differentiable_in u,1 ; ::_thesis: f is_hpartial_differentiable`31_in u
then consider x0, y0, z0 being Real such that
A2: ( 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)) ) ) by PDIFF_4:13;
thus f is_hpartial_differentiable`31_in u by A2, Def7; ::_thesis: verum
end;
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 )
proof
let u be Element of REAL 3; ::_thesis: 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 )
let f be PartFunc of (REAL 3),REAL; ::_thesis: ( f is_hpartial_differentiable`32_in u iff pdiff1 (f,3) is_partial_differentiable_in u,2 )
thus ( f is_hpartial_differentiable`32_in u implies pdiff1 (f,3) is_partial_differentiable_in u,2 ) ::_thesis: ( pdiff1 (f,3) is_partial_differentiable_in u,2 implies f is_hpartial_differentiable`32_in u )
proof
assume f is_hpartial_differentiable`32_in u ; ::_thesis: pdiff1 (f,3) is_partial_differentiable_in u,2
then consider x0, y0, z0 being Real such that
A1: ( 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)) ) ) by Def8;
thus pdiff1 (f,3) is_partial_differentiable_in u,2 by A1, PDIFF_4:14; ::_thesis: verum
end;
assume pdiff1 (f,3) is_partial_differentiable_in u,2 ; ::_thesis: f is_hpartial_differentiable`32_in u
then consider x0, y0, z0 being Real such that
A2: ( 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)) ) ) by PDIFF_4:14;
thus f is_hpartial_differentiable`32_in u by A2, Def8; ::_thesis: verum
end;
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 )
proof
let u be Element of REAL 3; ::_thesis: 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 )
let f be PartFunc of (REAL 3),REAL; ::_thesis: ( f is_hpartial_differentiable`33_in u iff pdiff1 (f,3) is_partial_differentiable_in u,3 )
thus ( f is_hpartial_differentiable`33_in u implies pdiff1 (f,3) is_partial_differentiable_in u,3 ) ::_thesis: ( pdiff1 (f,3) is_partial_differentiable_in u,3 implies f is_hpartial_differentiable`33_in u )
proof
assume f is_hpartial_differentiable`33_in u ; ::_thesis: pdiff1 (f,3) is_partial_differentiable_in u,3
then consider x0, y0, z0 being Real such that
A1: ( 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)) ) ) by Def9;
thus pdiff1 (f,3) is_partial_differentiable_in u,3 by A1, PDIFF_4:15; ::_thesis: verum
end;
assume pdiff1 (f,3) is_partial_differentiable_in u,3 ; ::_thesis: f is_hpartial_differentiable`33_in u
then consider x0, y0, z0 being Real such that
A2: ( 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)) ) ) by PDIFF_4:15;
thus f is_hpartial_differentiable`33_in u by A2, Def9; ::_thesis: verum
end;
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
let u be Element of REAL 3; ::_thesis: 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)
let f be PartFunc of (REAL 3),REAL; ::_thesis: ( f is_hpartial_differentiable`11_in u implies hpartdiff11 (f,u) = partdiff ((pdiff1 (f,1)),u,1) )
assume A1: f is_hpartial_differentiable`11_in u ; ::_thesis: hpartdiff11 (f,u) = partdiff ((pdiff1 (f,1)),u,1)
consider x0, y0, z0 being Real such that
A2: u = <*x0,y0,z0*> by FINSEQ_2:103;
hpartdiff11 (f,u) = diff ((SVF1 (1,(pdiff1 (f,1)),u)),x0) by A1, A2, Th10
.= partdiff ((pdiff1 (f,1)),u,1) by A2, PDIFF_4:19 ;
hence hpartdiff11 (f,u) = partdiff ((pdiff1 (f,1)),u,1) ; ::_thesis: verum
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
let u be Element of REAL 3; ::_thesis: 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)
let f be PartFunc of (REAL 3),REAL; ::_thesis: ( f is_hpartial_differentiable`12_in u implies hpartdiff12 (f,u) = partdiff ((pdiff1 (f,1)),u,2) )
assume A1: f is_hpartial_differentiable`12_in u ; ::_thesis: hpartdiff12 (f,u) = partdiff ((pdiff1 (f,1)),u,2)
consider x0, y0, z0 being Real such that
A2: u = <*x0,y0,z0*> by FINSEQ_2:103;
hpartdiff12 (f,u) = diff ((SVF1 (2,(pdiff1 (f,1)),u)),y0) by A1, A2, Th11
.= partdiff ((pdiff1 (f,1)),u,2) by A2, PDIFF_4:20 ;
hence hpartdiff12 (f,u) = partdiff ((pdiff1 (f,1)),u,2) ; ::_thesis: verum
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
let u be Element of REAL 3; ::_thesis: 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)
let f be PartFunc of (REAL 3),REAL; ::_thesis: ( f is_hpartial_differentiable`13_in u implies hpartdiff13 (f,u) = partdiff ((pdiff1 (f,1)),u,3) )
assume A1: f is_hpartial_differentiable`13_in u ; ::_thesis: hpartdiff13 (f,u) = partdiff ((pdiff1 (f,1)),u,3)
consider x0, y0, z0 being Real such that
A2: u = <*x0,y0,z0*> by FINSEQ_2:103;
hpartdiff13 (f,u) = diff ((SVF1 (3,(pdiff1 (f,1)),u)),z0) by A1, A2, Th12
.= partdiff ((pdiff1 (f,1)),u,3) by A2, PDIFF_4:21 ;
hence hpartdiff13 (f,u) = partdiff ((pdiff1 (f,1)),u,3) ; ::_thesis: verum
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
let u be Element of REAL 3; ::_thesis: 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)
let f be PartFunc of (REAL 3),REAL; ::_thesis: ( f is_hpartial_differentiable`21_in u implies hpartdiff21 (f,u) = partdiff ((pdiff1 (f,2)),u,1) )
assume A1: f is_hpartial_differentiable`21_in u ; ::_thesis: hpartdiff21 (f,u) = partdiff ((pdiff1 (f,2)),u,1)
consider x0, y0, z0 being Real such that
A2: u = <*x0,y0,z0*> by FINSEQ_2:103;
hpartdiff21 (f,u) = diff ((SVF1 (1,(pdiff1 (f,2)),u)),x0) by A1, A2, Th13
.= partdiff ((pdiff1 (f,2)),u,1) by A2, PDIFF_4:19 ;
hence hpartdiff21 (f,u) = partdiff ((pdiff1 (f,2)),u,1) ; ::_thesis: verum
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
let u be Element of REAL 3; ::_thesis: 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)
let f be PartFunc of (REAL 3),REAL; ::_thesis: ( f is_hpartial_differentiable`22_in u implies hpartdiff22 (f,u) = partdiff ((pdiff1 (f,2)),u,2) )
assume A1: f is_hpartial_differentiable`22_in u ; ::_thesis: hpartdiff22 (f,u) = partdiff ((pdiff1 (f,2)),u,2)
consider x0, y0, z0 being Real such that
A2: u = <*x0,y0,z0*> by FINSEQ_2:103;
hpartdiff22 (f,u) = diff ((SVF1 (2,(pdiff1 (f,2)),u)),y0) by A1, A2, Th14
.= partdiff ((pdiff1 (f,2)),u,2) by A2, PDIFF_4:20 ;
hence hpartdiff22 (f,u) = partdiff ((pdiff1 (f,2)),u,2) ; ::_thesis: verum
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
let u be Element of REAL 3; ::_thesis: 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)
let f be PartFunc of (REAL 3),REAL; ::_thesis: ( f is_hpartial_differentiable`23_in u implies hpartdiff23 (f,u) = partdiff ((pdiff1 (f,2)),u,3) )
assume A1: f is_hpartial_differentiable`23_in u ; ::_thesis: hpartdiff23 (f,u) = partdiff ((pdiff1 (f,2)),u,3)
consider x0, y0, z0 being Real such that
A2: u = <*x0,y0,z0*> by FINSEQ_2:103;
hpartdiff23 (f,u) = diff ((SVF1 (3,(pdiff1 (f,2)),u)),z0) by A1, A2, Th15
.= partdiff ((pdiff1 (f,2)),u,3) by A2, PDIFF_4:21 ;
hence hpartdiff23 (f,u) = partdiff ((pdiff1 (f,2)),u,3) ; ::_thesis: verum
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
let u be Element of REAL 3; ::_thesis: 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)
let f be PartFunc of (REAL 3),REAL; ::_thesis: ( f is_hpartial_differentiable`31_in u implies hpartdiff31 (f,u) = partdiff ((pdiff1 (f,3)),u,1) )
assume A1: f is_hpartial_differentiable`31_in u ; ::_thesis: hpartdiff31 (f,u) = partdiff ((pdiff1 (f,3)),u,1)
consider x0, y0, z0 being Real such that
A2: u = <*x0,y0,z0*> by FINSEQ_2:103;
hpartdiff31 (f,u) = diff ((SVF1 (1,(pdiff1 (f,3)),u)),x0) by A1, A2, Th16
.= partdiff ((pdiff1 (f,3)),u,1) by A2, PDIFF_4:19 ;
hence hpartdiff31 (f,u) = partdiff ((pdiff1 (f,3)),u,1) ; ::_thesis: verum
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
let u be Element of REAL 3; ::_thesis: 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)
let f be PartFunc of (REAL 3),REAL; ::_thesis: ( f is_hpartial_differentiable`32_in u implies hpartdiff32 (f,u) = partdiff ((pdiff1 (f,3)),u,2) )
assume A1: f is_hpartial_differentiable`32_in u ; ::_thesis: hpartdiff32 (f,u) = partdiff ((pdiff1 (f,3)),u,2)
consider x0, y0, z0 being Real such that
A2: u = <*x0,y0,z0*> by FINSEQ_2:103;
hpartdiff32 (f,u) = diff ((SVF1 (2,(pdiff1 (f,3)),u)),y0) by A1, A2, Th17
.= partdiff ((pdiff1 (f,3)),u,2) by A2, PDIFF_4:20 ;
hence hpartdiff32 (f,u) = partdiff ((pdiff1 (f,3)),u,2) ; ::_thesis: verum
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
let u be Element of REAL 3; ::_thesis: 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)
let f be PartFunc of (REAL 3),REAL; ::_thesis: ( f is_hpartial_differentiable`33_in u implies hpartdiff33 (f,u) = partdiff ((pdiff1 (f,3)),u,3) )
assume A1: f is_hpartial_differentiable`33_in u ; ::_thesis: hpartdiff33 (f,u) = partdiff ((pdiff1 (f,3)),u,3)
consider x0, y0, z0 being Real such that
A2: u = <*x0,y0,z0*> by FINSEQ_2:103;
hpartdiff33 (f,u) = diff ((SVF1 (3,(pdiff1 (f,3)),u)),z0) by A1, A2, Th18
.= partdiff ((pdiff1 (f,3)),u,3) by A2, PDIFF_4:21 ;
hence hpartdiff33 (f,u) = partdiff ((pdiff1 (f,3)),u,3) ; ::_thesis: verum
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
let f be PartFunc of (REAL 3),REAL; ::_thesis: 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))) )
let u0 be Element of REAL 3; ::_thesis: 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))) )
let N be Neighbourhood of (proj (1,3)) . u0; ::_thesis: ( f is_hpartial_differentiable`11_in u0 & N c= dom (SVF1 (1,(pdiff1 (f,1)),u0)) implies 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))) ) )
assume A1: ( f is_hpartial_differentiable`11_in u0 & N c= dom (SVF1 (1,(pdiff1 (f,1)),u0)) ) ; ::_thesis: 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))) )
let h be non-zero 0 -convergent Real_Sequence; ::_thesis: 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))) )
let c be constant Real_Sequence; ::_thesis: ( rng c = {((proj (1,3)) . u0)} & rng (h + c) c= N implies ( (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))) ) )
assume A2: ( rng c = {((proj (1,3)) . u0)} & rng (h + c) c= N ) ; ::_thesis: ( (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))) )
A3: pdiff1 (f,1) is_partial_differentiable_in u0,1 by A1, Th19;
consider x0, y0, z0 being Real such that
A4: u0 = <*x0,y0,z0*> by FINSEQ_2:103;
partdiff ((pdiff1 (f,1)),u0,1) = diff ((SVF1 (1,(pdiff1 (f,1)),u0)),x0) by A4, PDIFF_4:19
.= hpartdiff11 (f,u0) by A1, A4, Th10 ;
hence ( (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))) ) by A1, A2, A3, PDIFF_4:25; ::_thesis: verum
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
let f be PartFunc of (REAL 3),REAL; ::_thesis: 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))) )
let u0 be Element of REAL 3; ::_thesis: 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))) )
let N be Neighbourhood of (proj (2,3)) . u0; ::_thesis: ( f is_hpartial_differentiable`12_in u0 & N c= dom (SVF1 (2,(pdiff1 (f,1)),u0)) implies 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))) ) )
assume A1: ( f is_hpartial_differentiable`12_in u0 & N c= dom (SVF1 (2,(pdiff1 (f,1)),u0)) ) ; ::_thesis: 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))) )
let h be non-zero 0 -convergent Real_Sequence; ::_thesis: 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))) )
let c be constant Real_Sequence; ::_thesis: ( rng c = {((proj (2,3)) . u0)} & rng (h + c) c= N implies ( (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))) ) )
assume A2: ( rng c = {((proj (2,3)) . u0)} & rng (h + c) c= N ) ; ::_thesis: ( (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))) )
A3: pdiff1 (f,1) is_partial_differentiable_in u0,2 by A1, Th20;
consider x0, y0, z0 being Real such that
A4: u0 = <*x0,y0,z0*> by FINSEQ_2:103;
partdiff ((pdiff1 (f,1)),u0,2) = diff ((SVF1 (2,(pdiff1 (f,1)),u0)),y0) by A4, PDIFF_4:20
.= hpartdiff12 (f,u0) by A1, A4, Th11 ;
hence ( (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))) ) by A1, A2, A3, PDIFF_4:26; ::_thesis: verum
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
let f be PartFunc of (REAL 3),REAL; ::_thesis: 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))) )
let u0 be Element of REAL 3; ::_thesis: 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))) )
let N be Neighbourhood of (proj (3,3)) . u0; ::_thesis: ( f is_hpartial_differentiable`13_in u0 & N c= dom (SVF1 (3,(pdiff1 (f,1)),u0)) implies 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))) ) )
assume A1: ( f is_hpartial_differentiable`13_in u0 & N c= dom (SVF1 (3,(pdiff1 (f,1)),u0)) ) ; ::_thesis: 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))) )
let h be non-zero 0 -convergent Real_Sequence; ::_thesis: 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))) )
let c be constant Real_Sequence; ::_thesis: ( rng c = {((proj (3,3)) . u0)} & rng (h + c) c= N implies ( (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))) ) )
assume A2: ( rng c = {((proj (3,3)) . u0)} & rng (h + c) c= N ) ; ::_thesis: ( (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))) )
A3: pdiff1 (f,1) is_partial_differentiable_in u0,3 by A1, Th21;
consider x0, y0, z0 being Real such that
A4: u0 = <*x0,y0,z0*> by FINSEQ_2:103;
partdiff ((pdiff1 (f,1)),u0,3) = diff ((SVF1 (3,(pdiff1 (f,1)),u0)),z0) by A4, PDIFF_4:21
.= hpartdiff13 (f,u0) by A1, A4, Th12 ;
hence ( (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))) ) by A1, A2, A3, PDIFF_4:27; ::_thesis: verum
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
let f be PartFunc of (REAL 3),REAL; ::_thesis: 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))) )
let u0 be Element of REAL 3; ::_thesis: 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))) )
let N be Neighbourhood of (proj (1,3)) . u0; ::_thesis: ( f is_hpartial_differentiable`21_in u0 & N c= dom (SVF1 (1,(pdiff1 (f,2)),u0)) implies 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))) ) )
assume A1: ( f is_hpartial_differentiable`21_in u0 & N c= dom (SVF1 (1,(pdiff1 (f,2)),u0)) ) ; ::_thesis: 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))) )
let h be non-zero 0 -convergent Real_Sequence; ::_thesis: 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))) )
let c be constant Real_Sequence; ::_thesis: ( rng c = {((proj (1,3)) . u0)} & rng (h + c) c= N implies ( (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))) ) )
assume A2: ( rng c = {((proj (1,3)) . u0)} & rng (h + c) c= N ) ; ::_thesis: ( (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))) )
A3: pdiff1 (f,2) is_partial_differentiable_in u0,1 by A1, Th22;
consider x0, y0, z0 being Real such that
A4: u0 = <*x0,y0,z0*> by FINSEQ_2:103;
partdiff ((pdiff1 (f,2)),u0,1) = diff ((SVF1 (1,(pdiff1 (f,2)),u0)),x0) by A4, PDIFF_4:19
.= hpartdiff21 (f,u0) by A1, A4, Th13 ;
hence ( (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))) ) by A1, A2, A3, PDIFF_4:25; ::_thesis: verum
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
let f be PartFunc of (REAL 3),REAL; ::_thesis: 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))) )
let u0 be Element of REAL 3; ::_thesis: 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))) )
let N be Neighbourhood of (proj (2,3)) . u0; ::_thesis: ( f is_hpartial_differentiable`22_in u0 & N c= dom (SVF1 (2,(pdiff1 (f,2)),u0)) implies 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))) ) )
assume A1: ( f is_hpartial_differentiable`22_in u0 & N c= dom (SVF1 (2,(pdiff1 (f,2)),u0)) ) ; ::_thesis: 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))) )
let h be non-zero 0 -convergent Real_Sequence; ::_thesis: 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))) )
let c be constant Real_Sequence; ::_thesis: ( rng c = {((proj (2,3)) . u0)} & rng (h + c) c= N implies ( (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))) ) )
assume A2: ( rng c = {((proj (2,3)) . u0)} & rng (h + c) c= N ) ; ::_thesis: ( (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))) )
A3: pdiff1 (f,2) is_partial_differentiable_in u0,2 by A1, Th23;
consider x0, y0, z0 being Real such that
A4: u0 = <*x0,y0,z0*> by FINSEQ_2:103;
partdiff ((pdiff1 (f,2)),u0,2) = diff ((SVF1 (2,(pdiff1 (f,2)),u0)),y0) by A4, PDIFF_4:20
.= hpartdiff22 (f,u0) by A1, A4, Th14 ;
hence ( (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))) ) by A1, A2, A3, PDIFF_4:26; ::_thesis: verum
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
let f be PartFunc of (REAL 3),REAL; ::_thesis: 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))) )
let u0 be Element of REAL 3; ::_thesis: 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))) )
let N be Neighbourhood of (proj (3,3)) . u0; ::_thesis: ( f is_hpartial_differentiable`23_in u0 & N c= dom (SVF1 (3,(pdiff1 (f,2)),u0)) implies 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))) ) )
assume A1: ( f is_hpartial_differentiable`23_in u0 & N c= dom (SVF1 (3,(pdiff1 (f,2)),u0)) ) ; ::_thesis: 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))) )
let h be non-zero 0 -convergent Real_Sequence; ::_thesis: 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))) )
let c be constant Real_Sequence; ::_thesis: ( rng c = {((proj (3,3)) . u0)} & rng (h + c) c= N implies ( (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))) ) )
assume A2: ( rng c = {((proj (3,3)) . u0)} & rng (h + c) c= N ) ; ::_thesis: ( (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))) )
A3: pdiff1 (f,2) is_partial_differentiable_in u0,3 by A1, Th24;
consider x0, y0, z0 being Real such that
A4: u0 = <*x0,y0,z0*> by FINSEQ_2:103;
partdiff ((pdiff1 (f,2)),u0,3) = diff ((SVF1 (3,(pdiff1 (f,2)),u0)),z0) by A4, PDIFF_4:21
.= hpartdiff23 (f,u0) by A1, A4, Th15 ;
hence ( (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))) ) by A1, A2, A3, PDIFF_4:27; ::_thesis: verum
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
let f be PartFunc of (REAL 3),REAL; ::_thesis: 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))) )
let u0 be Element of REAL 3; ::_thesis: 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))) )
let N be Neighbourhood of (proj (1,3)) . u0; ::_thesis: ( f is_hpartial_differentiable`31_in u0 & N c= dom (SVF1 (1,(pdiff1 (f,3)),u0)) implies 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))) ) )
assume A1: ( f is_hpartial_differentiable`31_in u0 & N c= dom (SVF1 (1,(pdiff1 (f,3)),u0)) ) ; ::_thesis: 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))) )
let h be non-zero 0 -convergent Real_Sequence; ::_thesis: 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))) )
let c be constant Real_Sequence; ::_thesis: ( rng c = {((proj (1,3)) . u0)} & rng (h + c) c= N implies ( (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))) ) )
assume A2: ( rng c = {((proj (1,3)) . u0)} & rng (h + c) c= N ) ; ::_thesis: ( (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))) )
A3: pdiff1 (f,3) is_partial_differentiable_in u0,1 by A1, Th25;
consider x0, y0, z0 being Real such that
A4: u0 = <*x0,y0,z0*> by FINSEQ_2:103;
partdiff ((pdiff1 (f,3)),u0,1) = diff ((SVF1 (1,(pdiff1 (f,3)),u0)),x0) by A4, PDIFF_4:19
.= hpartdiff31 (f,u0) by A1, A4, Th16 ;
hence ( (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))) ) by A1, A2, A3, PDIFF_4:25; ::_thesis: verum
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
let f be PartFunc of (REAL 3),REAL; ::_thesis: 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))) )
let u0 be Element of REAL 3; ::_thesis: 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))) )
let N be Neighbourhood of (proj (2,3)) . u0; ::_thesis: ( f is_hpartial_differentiable`32_in u0 & N c= dom (SVF1 (2,(pdiff1 (f,3)),u0)) implies 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))) ) )
assume A1: ( f is_hpartial_differentiable`32_in u0 & N c= dom (SVF1 (2,(pdiff1 (f,3)),u0)) ) ; ::_thesis: 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))) )
let h be non-zero 0 -convergent Real_Sequence; ::_thesis: 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))) )
let c be constant Real_Sequence; ::_thesis: ( rng c = {((proj (2,3)) . u0)} & rng (h + c) c= N implies ( (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))) ) )
assume A2: ( rng c = {((proj (2,3)) . u0)} & rng (h + c) c= N ) ; ::_thesis: ( (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))) )
A3: pdiff1 (f,3) is_partial_differentiable_in u0,2 by A1, Th26;
consider x0, y0, z0 being Real such that
A4: u0 = <*x0,y0,z0*> by FINSEQ_2:103;
partdiff ((pdiff1 (f,3)),u0,2) = diff ((SVF1 (2,(pdiff1 (f,3)),u0)),y0) by A4, PDIFF_4:20
.= hpartdiff32 (f,u0) by A1, A4, Th17 ;
hence ( (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))) ) by A1, A2, A3, PDIFF_4:26; ::_thesis: verum
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
let f be PartFunc of (REAL 3),REAL; ::_thesis: 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))) )
let u0 be Element of REAL 3; ::_thesis: 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))) )
let N be Neighbourhood of (proj (3,3)) . u0; ::_thesis: ( f is_hpartial_differentiable`33_in u0 & N c= dom (SVF1 (3,(pdiff1 (f,3)),u0)) implies 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))) ) )
assume A1: ( f is_hpartial_differentiable`33_in u0 & N c= dom (SVF1 (3,(pdiff1 (f,3)),u0)) ) ; ::_thesis: 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))) )
let h be non-zero 0 -convergent Real_Sequence; ::_thesis: 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))) )
let c be constant Real_Sequence; ::_thesis: ( rng c = {((proj (3,3)) . u0)} & rng (h + c) c= N implies ( (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))) ) )
assume A2: ( rng c = {((proj (3,3)) . u0)} & rng (h + c) c= N ) ; ::_thesis: ( (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))) )
A3: pdiff1 (f,3) is_partial_differentiable_in u0,3 by A1, Th27;
consider x0, y0, z0 being Real such that
A4: u0 = <*x0,y0,z0*> by FINSEQ_2:103;
partdiff ((pdiff1 (f,3)),u0,3) = diff ((SVF1 (3,(pdiff1 (f,3)),u0)),z0) by A4, PDIFF_4:21
.= hpartdiff33 (f,u0) by A1, A4, Th18 ;
hence ( (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))) ) by A1, A2, A3, PDIFF_4:27; ::_thesis: verum
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
let u0 be Element of REAL 3; ::_thesis: 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)) )
let f1, f2 be PartFunc of (REAL 3),REAL; ::_thesis: ( f1 is_hpartial_differentiable`11_in u0 & f2 is_hpartial_differentiable`11_in u0 implies ( (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)) ) )
assume that
A1: f1 is_hpartial_differentiable`11_in u0 and
A2: f2 is_hpartial_differentiable`11_in u0 ; ::_thesis: ( (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)) )
A3: pdiff1 (f1,1) is_partial_differentiable_in u0,1 by A1, Th19;
A4: pdiff1 (f2,1) is_partial_differentiable_in u0,1 by A2, Th19;
then ( (pdiff1 (f1,1)) + (pdiff1 (f2,1)) is_partial_differentiable_in u0,1 & partdiff (((pdiff1 (f1,1)) + (pdiff1 (f2,1))),u0,1) = (partdiff ((pdiff1 (f1,1)),u0,1)) + (partdiff ((pdiff1 (f2,1)),u0,1)) ) by A3, PDIFF_1:29;
then partdiff (((pdiff1 (f1,1)) + (pdiff1 (f2,1))),u0,1) = (hpartdiff11 (f1,u0)) + (partdiff ((pdiff1 (f2,1)),u0,1)) by A1, Th28
.= (hpartdiff11 (f1,u0)) + (hpartdiff11 (f2,u0)) by A2, Th28 ;
hence ( (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)) ) by A3, A4, PDIFF_1:29; ::_thesis: verum
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
let u0 be Element of REAL 3; ::_thesis: 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)) )
let f1, f2 be PartFunc of (REAL 3),REAL; ::_thesis: ( f1 is_hpartial_differentiable`12_in u0 & f2 is_hpartial_differentiable`12_in u0 implies ( (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)) ) )
assume that
A1: f1 is_hpartial_differentiable`12_in u0 and
A2: f2 is_hpartial_differentiable`12_in u0 ; ::_thesis: ( (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)) )
A3: pdiff1 (f1,1) is_partial_differentiable_in u0,2 by A1, Th20;
A4: pdiff1 (f2,1) is_partial_differentiable_in u0,2 by A2, Th20;
then ( (pdiff1 (f1,1)) + (pdiff1 (f2,1)) is_partial_differentiable_in u0,2 & partdiff (((pdiff1 (f1,1)) + (pdiff1 (f2,1))),u0,2) = (partdiff ((pdiff1 (f1,1)),u0,2)) + (partdiff ((pdiff1 (f2,1)),u0,2)) ) by A3, PDIFF_1:29;
then partdiff (((pdiff1 (f1,1)) + (pdiff1 (f2,1))),u0,2) = (hpartdiff12 (f1,u0)) + (partdiff ((pdiff1 (f2,1)),u0,2)) by A1, Th29
.= (hpartdiff12 (f1,u0)) + (hpartdiff12 (f2,u0)) by A2, Th29 ;
hence ( (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)) ) by A3, A4, PDIFF_1:29; ::_thesis: verum
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
let u0 be Element of REAL 3; ::_thesis: 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)) )
let f1, f2 be PartFunc of (REAL 3),REAL; ::_thesis: ( f1 is_hpartial_differentiable`13_in u0 & f2 is_hpartial_differentiable`13_in u0 implies ( (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)) ) )
assume that
A1: f1 is_hpartial_differentiable`13_in u0 and
A2: f2 is_hpartial_differentiable`13_in u0 ; ::_thesis: ( (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)) )
A3: pdiff1 (f1,1) is_partial_differentiable_in u0,3 by A1, Th21;
A4: pdiff1 (f2,1) is_partial_differentiable_in u0,3 by A2, Th21;
then ( (pdiff1 (f1,1)) + (pdiff1 (f2,1)) is_partial_differentiable_in u0,3 & partdiff (((pdiff1 (f1,1)) + (pdiff1 (f2,1))),u0,3) = (partdiff ((pdiff1 (f1,1)),u0,3)) + (partdiff ((pdiff1 (f2,1)),u0,3)) ) by A3, PDIFF_1:29;
then partdiff (((pdiff1 (f1,1)) + (pdiff1 (f2,1))),u0,3) = (hpartdiff13 (f1,u0)) + (partdiff ((pdiff1 (f2,1)),u0,3)) by A1, Th30
.= (hpartdiff13 (f1,u0)) + (hpartdiff13 (f2,u0)) by A2, Th30 ;
hence ( (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)) ) by A3, A4, PDIFF_1:29; ::_thesis: verum
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
let u0 be Element of REAL 3; ::_thesis: 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)) )
let f1, f2 be PartFunc of (REAL 3),REAL; ::_thesis: ( f1 is_hpartial_differentiable`21_in u0 & f2 is_hpartial_differentiable`21_in u0 implies ( (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)) ) )
assume that
A1: f1 is_hpartial_differentiable`21_in u0 and
A2: f2 is_hpartial_differentiable`21_in u0 ; ::_thesis: ( (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)) )
A3: pdiff1 (f1,2) is_partial_differentiable_in u0,1 by A1, Th22;
A4: pdiff1 (f2,2) is_partial_differentiable_in u0,1 by A2, Th22;
then ( (pdiff1 (f1,2)) + (pdiff1 (f2,2)) is_partial_differentiable_in u0,1 & partdiff (((pdiff1 (f1,2)) + (pdiff1 (f2,2))),u0,1) = (partdiff ((pdiff1 (f1,2)),u0,1)) + (partdiff ((pdiff1 (f2,2)),u0,1)) ) by A3, PDIFF_1:29;
then partdiff (((pdiff1 (f1,2)) + (pdiff1 (f2,2))),u0,1) = (hpartdiff21 (f1,u0)) + (partdiff ((pdiff1 (f2,2)),u0,1)) by A1, Th31
.= (hpartdiff21 (f1,u0)) + (hpartdiff21 (f2,u0)) by A2, Th31 ;
hence ( (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)) ) by A3, A4, PDIFF_1:29; ::_thesis: verum
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
let u0 be Element of REAL 3; ::_thesis: 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)) )
let f1, f2 be PartFunc of (REAL 3),REAL; ::_thesis: ( f1 is_hpartial_differentiable`22_in u0 & f2 is_hpartial_differentiable`22_in u0 implies ( (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)) ) )
assume that
A1: f1 is_hpartial_differentiable`22_in u0 and
A2: f2 is_hpartial_differentiable`22_in u0 ; ::_thesis: ( (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)) )
A3: pdiff1 (f1,2) is_partial_differentiable_in u0,2 by A1, Th23;
A4: pdiff1 (f2,2) is_partial_differentiable_in u0,2 by A2, Th23;
then ( (pdiff1 (f1,2)) + (pdiff1 (f2,2)) is_partial_differentiable_in u0,2 & partdiff (((pdiff1 (f1,2)) + (pdiff1 (f2,2))),u0,2) = (partdiff ((pdiff1 (f1,2)),u0,2)) + (partdiff ((pdiff1 (f2,2)),u0,2)) ) by A3, PDIFF_1:29;
then partdiff (((pdiff1 (f1,2)) + (pdiff1 (f2,2))),u0,2) = (hpartdiff22 (f1,u0)) + (partdiff ((pdiff1 (f2,2)),u0,2)) by A1, Th32
.= (hpartdiff22 (f1,u0)) + (hpartdiff22 (f2,u0)) by A2, Th32 ;
hence ( (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)) ) by A3, A4, PDIFF_1:29; ::_thesis: verum
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
let u0 be Element of REAL 3; ::_thesis: 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)) )
let f1, f2 be PartFunc of (REAL 3),REAL; ::_thesis: ( f1 is_hpartial_differentiable`23_in u0 & f2 is_hpartial_differentiable`23_in u0 implies ( (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)) ) )
assume that
A1: f1 is_hpartial_differentiable`23_in u0 and
A2: f2 is_hpartial_differentiable`23_in u0 ; ::_thesis: ( (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)) )
A3: pdiff1 (f1,2) is_partial_differentiable_in u0,3 by A1, Th24;
A4: pdiff1 (f2,2) is_partial_differentiable_in u0,3 by A2, Th24;
then ( (pdiff1 (f1,2)) + (pdiff1 (f2,2)) is_partial_differentiable_in u0,3 & partdiff (((pdiff1 (f1,2)) + (pdiff1 (f2,2))),u0,3) = (partdiff ((pdiff1 (f1,2)),u0,3)) + (partdiff ((pdiff1 (f2,2)),u0,3)) ) by A3, PDIFF_1:29;
then partdiff (((pdiff1 (f1,2)) + (pdiff1 (f2,2))),u0,3) = (hpartdiff23 (f1,u0)) + (partdiff ((pdiff1 (f2,2)),u0,3)) by A1, Th33
.= (hpartdiff23 (f1,u0)) + (hpartdiff23 (f2,u0)) by A2, Th33 ;
hence ( (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)) ) by A3, A4, PDIFF_1:29; ::_thesis: verum
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
let u0 be Element of REAL 3; ::_thesis: 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)) )
let f1, f2 be PartFunc of (REAL 3),REAL; ::_thesis: ( f1 is_hpartial_differentiable`11_in u0 & f2 is_hpartial_differentiable`11_in u0 implies ( (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)) ) )
assume that
A1: f1 is_hpartial_differentiable`11_in u0 and
A2: f2 is_hpartial_differentiable`11_in u0 ; ::_thesis: ( (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)) )
A3: pdiff1 (f1,1) is_partial_differentiable_in u0,1 by A1, Th19;
A4: pdiff1 (f2,1) is_partial_differentiable_in u0,1 by A2, Th19;
then ( (pdiff1 (f1,1)) - (pdiff1 (f2,1)) is_partial_differentiable_in u0,1 & partdiff (((pdiff1 (f1,1)) - (pdiff1 (f2,1))),u0,1) = (partdiff ((pdiff1 (f1,1)),u0,1)) - (partdiff ((pdiff1 (f2,1)),u0,1)) ) by A3, PDIFF_1:31;
then partdiff (((pdiff1 (f1,1)) - (pdiff1 (f2,1))),u0,1) = (hpartdiff11 (f1,u0)) - (partdiff ((pdiff1 (f2,1)),u0,1)) by A1, Th28
.= (hpartdiff11 (f1,u0)) - (hpartdiff11 (f2,u0)) by A2, Th28 ;
hence ( (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)) ) by A3, A4, PDIFF_1:31; ::_thesis: verum
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
let u0 be Element of REAL 3; ::_thesis: 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)) )
let f1, f2 be PartFunc of (REAL 3),REAL; ::_thesis: ( f1 is_hpartial_differentiable`12_in u0 & f2 is_hpartial_differentiable`12_in u0 implies ( (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)) ) )
assume that
A1: f1 is_hpartial_differentiable`12_in u0 and
A2: f2 is_hpartial_differentiable`12_in u0 ; ::_thesis: ( (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)) )
A3: pdiff1 (f1,1) is_partial_differentiable_in u0,2 by A1, Th20;
A4: pdiff1 (f2,1) is_partial_differentiable_in u0,2 by A2, Th20;
then ( (pdiff1 (f1,1)) - (pdiff1 (f2,1)) is_partial_differentiable_in u0,2 & partdiff (((pdiff1 (f1,1)) - (pdiff1 (f2,1))),u0,2) = (partdiff ((pdiff1 (f1,1)),u0,2)) - (partdiff ((pdiff1 (f2,1)),u0,2)) ) by A3, PDIFF_1:31;
then partdiff (((pdiff1 (f1,1)) - (pdiff1 (f2,1))),u0,2) = (hpartdiff12 (f1,u0)) - (partdiff ((pdiff1 (f2,1)),u0,2)) by A1, Th29
.= (hpartdiff12 (f1,u0)) - (hpartdiff12 (f2,u0)) by A2, Th29 ;
hence ( (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)) ) by A3, A4, PDIFF_1:31; ::_thesis: verum
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
let u0 be Element of REAL 3; ::_thesis: 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)) )
let f1, f2 be PartFunc of (REAL 3),REAL; ::_thesis: ( f1 is_hpartial_differentiable`13_in u0 & f2 is_hpartial_differentiable`13_in u0 implies ( (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)) ) )
assume that
A1: f1 is_hpartial_differentiable`13_in u0 and
A2: f2 is_hpartial_differentiable`13_in u0 ; ::_thesis: ( (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)) )
A3: pdiff1 (f1,1) is_partial_differentiable_in u0,3 by A1, Th21;
A4: pdiff1 (f2,1) is_partial_differentiable_in u0,3 by A2, Th21;
then ( (pdiff1 (f1,1)) - (pdiff1 (f2,1)) is_partial_differentiable_in u0,3 & partdiff (((pdiff1 (f1,1)) - (pdiff1 (f2,1))),u0,3) = (partdiff ((pdiff1 (f1,1)),u0,3)) - (partdiff ((pdiff1 (f2,1)),u0,3)) ) by A3, PDIFF_1:31;
then partdiff (((pdiff1 (f1,1)) - (pdiff1 (f2,1))),u0,3) = (hpartdiff13 (f1,u0)) - (partdiff ((pdiff1 (f2,1)),u0,3)) by A1, Th30
.= (hpartdiff13 (f1,u0)) - (hpartdiff13 (f2,u0)) by A2, Th30 ;
hence ( (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)) ) by A3, A4, PDIFF_1:31; ::_thesis: verum
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
let u0 be Element of REAL 3; ::_thesis: 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)) )
let f1, f2 be PartFunc of (REAL 3),REAL; ::_thesis: ( f1 is_hpartial_differentiable`21_in u0 & f2 is_hpartial_differentiable`21_in u0 implies ( (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)) ) )
assume that
A1: f1 is_hpartial_differentiable`21_in u0 and
A2: f2 is_hpartial_differentiable`21_in u0 ; ::_thesis: ( (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)) )
A3: pdiff1 (f1,2) is_partial_differentiable_in u0,1 by A1, Th22;
A4: pdiff1 (f2,2) is_partial_differentiable_in u0,1 by A2, Th22;
then ( (pdiff1 (f1,2)) - (pdiff1 (f2,2)) is_partial_differentiable_in u0,1 & partdiff (((pdiff1 (f1,2)) - (pdiff1 (f2,2))),u0,1) = (partdiff ((pdiff1 (f1,2)),u0,1)) - (partdiff ((pdiff1 (f2,2)),u0,1)) ) by A3, PDIFF_1:31;
then partdiff (((pdiff1 (f1,2)) - (pdiff1 (f2,2))),u0,1) = (hpartdiff21 (f1,u0)) - (partdiff ((pdiff1 (f2,2)),u0,1)) by A1, Th31
.= (hpartdiff21 (f1,u0)) - (hpartdiff21 (f2,u0)) by A2, Th31 ;
hence ( (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)) ) by A3, A4, PDIFF_1:31; ::_thesis: verum
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
let u0 be Element of REAL 3; ::_thesis: 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)) )
let f1, f2 be PartFunc of (REAL 3),REAL; ::_thesis: ( f1 is_hpartial_differentiable`22_in u0 & f2 is_hpartial_differentiable`22_in u0 implies ( (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)) ) )
assume that
A1: f1 is_hpartial_differentiable`22_in u0 and
A2: f2 is_hpartial_differentiable`22_in u0 ; ::_thesis: ( (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)) )
A3: pdiff1 (f1,2) is_partial_differentiable_in u0,2 by A1, Th23;
A4: pdiff1 (f2,2) is_partial_differentiable_in u0,2 by A2, Th23;
then ( (pdiff1 (f1,2)) - (pdiff1 (f2,2)) is_partial_differentiable_in u0,2 & partdiff (((pdiff1 (f1,2)) - (pdiff1 (f2,2))),u0,2) = (partdiff ((pdiff1 (f1,2)),u0,2)) - (partdiff ((pdiff1 (f2,2)),u0,2)) ) by A3, PDIFF_1:31;
then partdiff (((pdiff1 (f1,2)) - (pdiff1 (f2,2))),u0,2) = (hpartdiff22 (f1,u0)) - (partdiff ((pdiff1 (f2,2)),u0,2)) by A1, Th32
.= (hpartdiff22 (f1,u0)) - (hpartdiff22 (f2,u0)) by A2, Th32 ;
hence ( (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)) ) by A3, A4, PDIFF_1:31; ::_thesis: verum
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
let u0 be Element of REAL 3; ::_thesis: 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)) )
let f1, f2 be PartFunc of (REAL 3),REAL; ::_thesis: ( f1 is_hpartial_differentiable`23_in u0 & f2 is_hpartial_differentiable`23_in u0 implies ( (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)) ) )
assume that
A1: f1 is_hpartial_differentiable`23_in u0 and
A2: f2 is_hpartial_differentiable`23_in u0 ; ::_thesis: ( (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)) )
A3: pdiff1 (f1,2) is_partial_differentiable_in u0,3 by A1, Th24;
A4: pdiff1 (f2,2) is_partial_differentiable_in u0,3 by A2, Th24;
then ( (pdiff1 (f1,2)) - (pdiff1 (f2,2)) is_partial_differentiable_in u0,3 & partdiff (((pdiff1 (f1,2)) - (pdiff1 (f2,2))),u0,3) = (partdiff ((pdiff1 (f1,2)),u0,3)) - (partdiff ((pdiff1 (f2,2)),u0,3)) ) by A3, PDIFF_1:31;
then partdiff (((pdiff1 (f1,2)) - (pdiff1 (f2,2))),u0,3) = (hpartdiff23 (f1,u0)) - (partdiff ((pdiff1 (f2,2)),u0,3)) by A1, Th33
.= (hpartdiff23 (f1,u0)) - (hpartdiff23 (f2,u0)) by A2, Th33 ;
hence ( (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)) ) by A3, A4, PDIFF_1:31; ::_thesis: verum
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
let r be Real; ::_thesis: 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)) )
let u0 be Element of REAL 3; ::_thesis: 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)) )
let f be PartFunc of (REAL 3),REAL; ::_thesis: ( f is_hpartial_differentiable`11_in u0 implies ( r (#) (pdiff1 (f,1)) is_partial_differentiable_in u0,1 & partdiff ((r (#) (pdiff1 (f,1))),u0,1) = r * (hpartdiff11 (f,u0)) ) )
assume A1: f is_hpartial_differentiable`11_in u0 ; ::_thesis: ( r (#) (pdiff1 (f,1)) is_partial_differentiable_in u0,1 & partdiff ((r (#) (pdiff1 (f,1))),u0,1) = r * (hpartdiff11 (f,u0)) )
then pdiff1 (f,1) is_partial_differentiable_in u0,1 by Th19;
then ( r (#) (pdiff1 (f,1)) is_partial_differentiable_in u0,1 & partdiff ((r (#) (pdiff1 (f,1))),u0,1) = r * (partdiff ((pdiff1 (f,1)),u0,1)) ) by PDIFF_1:33;
hence ( r (#) (pdiff1 (f,1)) is_partial_differentiable_in u0,1 & partdiff ((r (#) (pdiff1 (f,1))),u0,1) = r * (hpartdiff11 (f,u0)) ) by A1, Th28; ::_thesis: verum
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
let r be Real; ::_thesis: 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)) )
let u0 be Element of REAL 3; ::_thesis: 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)) )
let f be PartFunc of (REAL 3),REAL; ::_thesis: ( f is_hpartial_differentiable`12_in u0 implies ( r (#) (pdiff1 (f,1)) is_partial_differentiable_in u0,2 & partdiff ((r (#) (pdiff1 (f,1))),u0,2) = r * (hpartdiff12 (f,u0)) ) )
assume A1: f is_hpartial_differentiable`12_in u0 ; ::_thesis: ( r (#) (pdiff1 (f,1)) is_partial_differentiable_in u0,2 & partdiff ((r (#) (pdiff1 (f,1))),u0,2) = r * (hpartdiff12 (f,u0)) )
then pdiff1 (f,1) is_partial_differentiable_in u0,2 by Th20;
then ( r (#) (pdiff1 (f,1)) is_partial_differentiable_in u0,2 & partdiff ((r (#) (pdiff1 (f,1))),u0,2) = r * (partdiff ((pdiff1 (f,1)),u0,2)) ) by PDIFF_1:33;
hence ( r (#) (pdiff1 (f,1)) is_partial_differentiable_in u0,2 & partdiff ((r (#) (pdiff1 (f,1))),u0,2) = r * (hpartdiff12 (f,u0)) ) by A1, Th29; ::_thesis: verum
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
let r be Real; ::_thesis: 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)) )
let u0 be Element of REAL 3; ::_thesis: 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)) )
let f be PartFunc of (REAL 3),REAL; ::_thesis: ( f is_hpartial_differentiable`13_in u0 implies ( r (#) (pdiff1 (f,1)) is_partial_differentiable_in u0,3 & partdiff ((r (#) (pdiff1 (f,1))),u0,3) = r * (hpartdiff13 (f,u0)) ) )
assume A1: f is_hpartial_differentiable`13_in u0 ; ::_thesis: ( r (#) (pdiff1 (f,1)) is_partial_differentiable_in u0,3 & partdiff ((r (#) (pdiff1 (f,1))),u0,3) = r * (hpartdiff13 (f,u0)) )
then pdiff1 (f,1) is_partial_differentiable_in u0,3 by Th21;
then ( r (#) (pdiff1 (f,1)) is_partial_differentiable_in u0,3 & partdiff ((r (#) (pdiff1 (f,1))),u0,3) = r * (partdiff ((pdiff1 (f,1)),u0,3)) ) by PDIFF_1:33;
hence ( r (#) (pdiff1 (f,1)) is_partial_differentiable_in u0,3 & partdiff ((r (#) (pdiff1 (f,1))),u0,3) = r * (hpartdiff13 (f,u0)) ) by A1, Th30; ::_thesis: verum
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
let r be Real; ::_thesis: 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)) )
let u0 be Element of REAL 3; ::_thesis: 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)) )
let f be PartFunc of (REAL 3),REAL; ::_thesis: ( f is_hpartial_differentiable`21_in u0 implies ( r (#) (pdiff1 (f,2)) is_partial_differentiable_in u0,1 & partdiff ((r (#) (pdiff1 (f,2))),u0,1) = r * (hpartdiff21 (f,u0)) ) )
assume A1: f is_hpartial_differentiable`21_in u0 ; ::_thesis: ( r (#) (pdiff1 (f,2)) is_partial_differentiable_in u0,1 & partdiff ((r (#) (pdiff1 (f,2))),u0,1) = r * (hpartdiff21 (f,u0)) )
then pdiff1 (f,2) is_partial_differentiable_in u0,1 by Th22;
then ( r (#) (pdiff1 (f,2)) is_partial_differentiable_in u0,1 & partdiff ((r (#) (pdiff1 (f,2))),u0,1) = r * (partdiff ((pdiff1 (f,2)),u0,1)) ) by PDIFF_1:33;
hence ( r (#) (pdiff1 (f,2)) is_partial_differentiable_in u0,1 & partdiff ((r (#) (pdiff1 (f,2))),u0,1) = r * (hpartdiff21 (f,u0)) ) by A1, Th31; ::_thesis: verum
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
let r be Real; ::_thesis: 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)) )
let u0 be Element of REAL 3; ::_thesis: 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)) )
let f be PartFunc of (REAL 3),REAL; ::_thesis: ( f is_hpartial_differentiable`22_in u0 implies ( r (#) (pdiff1 (f,2)) is_partial_differentiable_in u0,2 & partdiff ((r (#) (pdiff1 (f,2))),u0,2) = r * (hpartdiff22 (f,u0)) ) )
assume A1: f is_hpartial_differentiable`22_in u0 ; ::_thesis: ( r (#) (pdiff1 (f,2)) is_partial_differentiable_in u0,2 & partdiff ((r (#) (pdiff1 (f,2))),u0,2) = r * (hpartdiff22 (f,u0)) )
then pdiff1 (f,2) is_partial_differentiable_in u0,2 by Th23;
then ( r (#) (pdiff1 (f,2)) is_partial_differentiable_in u0,2 & partdiff ((r (#) (pdiff1 (f,2))),u0,2) = r * (partdiff ((pdiff1 (f,2)),u0,2)) ) by PDIFF_1:33;
hence ( r (#) (pdiff1 (f,2)) is_partial_differentiable_in u0,2 & partdiff ((r (#) (pdiff1 (f,2))),u0,2) = r * (hpartdiff22 (f,u0)) ) by A1, Th32; ::_thesis: verum
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
let r be Real; ::_thesis: 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)) )
let u0 be Element of REAL 3; ::_thesis: 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)) )
let f be PartFunc of (REAL 3),REAL; ::_thesis: ( f is_hpartial_differentiable`23_in u0 implies ( r (#) (pdiff1 (f,2)) is_partial_differentiable_in u0,3 & partdiff ((r (#) (pdiff1 (f,2))),u0,3) = r * (hpartdiff23 (f,u0)) ) )
assume A1: f is_hpartial_differentiable`23_in u0 ; ::_thesis: ( r (#) (pdiff1 (f,2)) is_partial_differentiable_in u0,3 & partdiff ((r (#) (pdiff1 (f,2))),u0,3) = r * (hpartdiff23 (f,u0)) )
then pdiff1 (f,2) is_partial_differentiable_in u0,3 by Th24;
then ( r (#) (pdiff1 (f,2)) is_partial_differentiable_in u0,3 & partdiff ((r (#) (pdiff1 (f,2))),u0,3) = r * (partdiff ((pdiff1 (f,2)),u0,3)) ) by PDIFF_1:33;
hence ( r (#) (pdiff1 (f,2)) is_partial_differentiable_in u0,3 & partdiff ((r (#) (pdiff1 (f,2))),u0,3) = r * (hpartdiff23 (f,u0)) ) by A1, Th33; ::_thesis: verum
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
let r be Real; ::_thesis: 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)) )
let u0 be Element of REAL 3; ::_thesis: 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)) )
let f be PartFunc of (REAL 3),REAL; ::_thesis: ( f is_hpartial_differentiable`31_in u0 implies ( r (#) (pdiff1 (f,3)) is_partial_differentiable_in u0,1 & partdiff ((r (#) (pdiff1 (f,3))),u0,1) = r * (hpartdiff31 (f,u0)) ) )
assume A1: f is_hpartial_differentiable`31_in u0 ; ::_thesis: ( r (#) (pdiff1 (f,3)) is_partial_differentiable_in u0,1 & partdiff ((r (#) (pdiff1 (f,3))),u0,1) = r * (hpartdiff31 (f,u0)) )
then pdiff1 (f,3) is_partial_differentiable_in u0,1 by Th25;
then ( r (#) (pdiff1 (f,3)) is_partial_differentiable_in u0,1 & partdiff ((r (#) (pdiff1 (f,3))),u0,1) = r * (partdiff ((pdiff1 (f,3)),u0,1)) ) by PDIFF_1:33;
hence ( r (#) (pdiff1 (f,3)) is_partial_differentiable_in u0,1 & partdiff ((r (#) (pdiff1 (f,3))),u0,1) = r * (hpartdiff31 (f,u0)) ) by A1, Th34; ::_thesis: verum
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
let r be Real; ::_thesis: 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)) )
let u0 be Element of REAL 3; ::_thesis: 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)) )
let f be PartFunc of (REAL 3),REAL; ::_thesis: ( f is_hpartial_differentiable`32_in u0 implies ( r (#) (pdiff1 (f,3)) is_partial_differentiable_in u0,2 & partdiff ((r (#) (pdiff1 (f,3))),u0,2) = r * (hpartdiff32 (f,u0)) ) )
assume A1: f is_hpartial_differentiable`32_in u0 ; ::_thesis: ( r (#) (pdiff1 (f,3)) is_partial_differentiable_in u0,2 & partdiff ((r (#) (pdiff1 (f,3))),u0,2) = r * (hpartdiff32 (f,u0)) )
then pdiff1 (f,3) is_partial_differentiable_in u0,2 by Th26;
then ( r (#) (pdiff1 (f,3)) is_partial_differentiable_in u0,2 & partdiff ((r (#) (pdiff1 (f,3))),u0,2) = r * (partdiff ((pdiff1 (f,3)),u0,2)) ) by PDIFF_1:33;
hence ( r (#) (pdiff1 (f,3)) is_partial_differentiable_in u0,2 & partdiff ((r (#) (pdiff1 (f,3))),u0,2) = r * (hpartdiff32 (f,u0)) ) by A1, Th35; ::_thesis: verum
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
let r be Real; ::_thesis: 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)) )
let u0 be Element of REAL 3; ::_thesis: 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)) )
let f be PartFunc of (REAL 3),REAL; ::_thesis: ( f is_hpartial_differentiable`33_in u0 implies ( r (#) (pdiff1 (f,3)) is_partial_differentiable_in u0,3 & partdiff ((r (#) (pdiff1 (f,3))),u0,3) = r * (hpartdiff33 (f,u0)) ) )
assume A1: f is_hpartial_differentiable`33_in u0 ; ::_thesis: ( r (#) (pdiff1 (f,3)) is_partial_differentiable_in u0,3 & partdiff ((r (#) (pdiff1 (f,3))),u0,3) = r * (hpartdiff33 (f,u0)) )
then pdiff1 (f,3) is_partial_differentiable_in u0,3 by Th27;
then ( r (#) (pdiff1 (f,3)) is_partial_differentiable_in u0,3 & partdiff ((r (#) (pdiff1 (f,3))),u0,3) = r * (partdiff ((pdiff1 (f,3)),u0,3)) ) by PDIFF_1:33;
hence ( r (#) (pdiff1 (f,3)) is_partial_differentiable_in u0,3 & partdiff ((r (#) (pdiff1 (f,3))),u0,3) = r * (hpartdiff33 (f,u0)) ) by A1, Th36; ::_thesis: verum
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
let u0 be Element of REAL 3; ::_thesis: 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
let f1, f2 be PartFunc of (REAL 3),REAL; ::_thesis: ( f1 is_hpartial_differentiable`11_in u0 & f2 is_hpartial_differentiable`11_in u0 implies (pdiff1 (f1,1)) (#) (pdiff1 (f2,1)) is_partial_differentiable_in u0,1 )
assume ( f1 is_hpartial_differentiable`11_in u0 & f2 is_hpartial_differentiable`11_in u0 ) ; ::_thesis: (pdiff1 (f1,1)) (#) (pdiff1 (f2,1)) is_partial_differentiable_in u0,1
then ( pdiff1 (f1,1) is_partial_differentiable_in u0,1 & pdiff1 (f2,1) is_partial_differentiable_in u0,1 ) by Th19;
hence (pdiff1 (f1,1)) (#) (pdiff1 (f2,1)) is_partial_differentiable_in u0,1 by PDIFF_4:28; ::_thesis: verum
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
let u0 be Element of REAL 3; ::_thesis: 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
let f1, f2 be PartFunc of (REAL 3),REAL; ::_thesis: ( f1 is_hpartial_differentiable`12_in u0 & f2 is_hpartial_differentiable`12_in u0 implies (pdiff1 (f1,1)) (#) (pdiff1 (f2,1)) is_partial_differentiable_in u0,2 )
assume ( f1 is_hpartial_differentiable`12_in u0 & f2 is_hpartial_differentiable`12_in u0 ) ; ::_thesis: (pdiff1 (f1,1)) (#) (pdiff1 (f2,1)) is_partial_differentiable_in u0,2
then ( pdiff1 (f1,1) is_partial_differentiable_in u0,2 & pdiff1 (f2,1) is_partial_differentiable_in u0,2 ) by Th20;
hence (pdiff1 (f1,1)) (#) (pdiff1 (f2,1)) is_partial_differentiable_in u0,2 by PDIFF_4:29; ::_thesis: verum
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
let u0 be Element of REAL 3; ::_thesis: 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
let f1, f2 be PartFunc of (REAL 3),REAL; ::_thesis: ( f1 is_hpartial_differentiable`13_in u0 & f2 is_hpartial_differentiable`13_in u0 implies (pdiff1 (f1,1)) (#) (pdiff1 (f2,1)) is_partial_differentiable_in u0,3 )
assume ( f1 is_hpartial_differentiable`13_in u0 & f2 is_hpartial_differentiable`13_in u0 ) ; ::_thesis: (pdiff1 (f1,1)) (#) (pdiff1 (f2,1)) is_partial_differentiable_in u0,3
then ( pdiff1 (f1,1) is_partial_differentiable_in u0,3 & pdiff1 (f2,1) is_partial_differentiable_in u0,3 ) by Th21;
hence (pdiff1 (f1,1)) (#) (pdiff1 (f2,1)) is_partial_differentiable_in u0,3 by PDIFF_4:30; ::_thesis: verum
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
let u0 be Element of REAL 3; ::_thesis: 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
let f1, f2 be PartFunc of (REAL 3),REAL; ::_thesis: ( f1 is_hpartial_differentiable`21_in u0 & f2 is_hpartial_differentiable`21_in u0 implies (pdiff1 (f1,2)) (#) (pdiff1 (f2,2)) is_partial_differentiable_in u0,1 )
assume ( f1 is_hpartial_differentiable`21_in u0 & f2 is_hpartial_differentiable`21_in u0 ) ; ::_thesis: (pdiff1 (f1,2)) (#) (pdiff1 (f2,2)) is_partial_differentiable_in u0,1
then ( pdiff1 (f1,2) is_partial_differentiable_in u0,1 & pdiff1 (f2,2) is_partial_differentiable_in u0,1 ) by Th22;
hence (pdiff1 (f1,2)) (#) (pdiff1 (f2,2)) is_partial_differentiable_in u0,1 by PDIFF_4:28; ::_thesis: verum
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
let u0 be Element of REAL 3; ::_thesis: 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
let f1, f2 be PartFunc of (REAL 3),REAL; ::_thesis: ( f1 is_hpartial_differentiable`22_in u0 & f2 is_hpartial_differentiable`22_in u0 implies (pdiff1 (f1,2)) (#) (pdiff1 (f2,2)) is_partial_differentiable_in u0,2 )
assume ( f1 is_hpartial_differentiable`22_in u0 & f2 is_hpartial_differentiable`22_in u0 ) ; ::_thesis: (pdiff1 (f1,2)) (#) (pdiff1 (f2,2)) is_partial_differentiable_in u0,2
then ( pdiff1 (f1,2) is_partial_differentiable_in u0,2 & pdiff1 (f2,2) is_partial_differentiable_in u0,2 ) by Th23;
hence (pdiff1 (f1,2)) (#) (pdiff1 (f2,2)) is_partial_differentiable_in u0,2 by PDIFF_4:29; ::_thesis: verum
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
let u0 be Element of REAL 3; ::_thesis: 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
let f1, f2 be PartFunc of (REAL 3),REAL; ::_thesis: ( f1 is_hpartial_differentiable`23_in u0 & f2 is_hpartial_differentiable`23_in u0 implies (pdiff1 (f1,2)) (#) (pdiff1 (f2,2)) is_partial_differentiable_in u0,3 )
assume ( f1 is_hpartial_differentiable`23_in u0 & f2 is_hpartial_differentiable`23_in u0 ) ; ::_thesis: (pdiff1 (f1,2)) (#) (pdiff1 (f2,2)) is_partial_differentiable_in u0,3
then ( pdiff1 (f1,2) is_partial_differentiable_in u0,3 & pdiff1 (f2,2) is_partial_differentiable_in u0,3 ) by Th24;
hence (pdiff1 (f1,2)) (#) (pdiff1 (f2,2)) is_partial_differentiable_in u0,3 by PDIFF_4:30; ::_thesis: verum
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
let u0 be Element of REAL 3; ::_thesis: 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
let f1, f2 be PartFunc of (REAL 3),REAL; ::_thesis: ( f1 is_hpartial_differentiable`31_in u0 & f2 is_hpartial_differentiable`31_in u0 implies (pdiff1 (f1,3)) (#) (pdiff1 (f2,3)) is_partial_differentiable_in u0,1 )
assume ( f1 is_hpartial_differentiable`31_in u0 & f2 is_hpartial_differentiable`31_in u0 ) ; ::_thesis: (pdiff1 (f1,3)) (#) (pdiff1 (f2,3)) is_partial_differentiable_in u0,1
then ( pdiff1 (f1,3) is_partial_differentiable_in u0,1 & pdiff1 (f2,3) is_partial_differentiable_in u0,1 ) by Th25;
hence (pdiff1 (f1,3)) (#) (pdiff1 (f2,3)) is_partial_differentiable_in u0,1 by PDIFF_4:28; ::_thesis: verum
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
let u0 be Element of REAL 3; ::_thesis: 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
let f1, f2 be PartFunc of (REAL 3),REAL; ::_thesis: ( f1 is_hpartial_differentiable`32_in u0 & f2 is_hpartial_differentiable`32_in u0 implies (pdiff1 (f1,3)) (#) (pdiff1 (f2,3)) is_partial_differentiable_in u0,2 )
assume ( f1 is_hpartial_differentiable`32_in u0 & f2 is_hpartial_differentiable`32_in u0 ) ; ::_thesis: (pdiff1 (f1,3)) (#) (pdiff1 (f2,3)) is_partial_differentiable_in u0,2
then ( pdiff1 (f1,3) is_partial_differentiable_in u0,2 & pdiff1 (f2,3) is_partial_differentiable_in u0,2 ) by Th26;
hence (pdiff1 (f1,3)) (#) (pdiff1 (f2,3)) is_partial_differentiable_in u0,2 by PDIFF_4:29; ::_thesis: verum
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
let u0 be Element of REAL 3; ::_thesis: 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
let f1, f2 be PartFunc of (REAL 3),REAL; ::_thesis: ( f1 is_hpartial_differentiable`33_in u0 & f2 is_hpartial_differentiable`33_in u0 implies (pdiff1 (f1,3)) (#) (pdiff1 (f2,3)) is_partial_differentiable_in u0,3 )
assume ( f1 is_hpartial_differentiable`33_in u0 & f2 is_hpartial_differentiable`33_in u0 ) ; ::_thesis: (pdiff1 (f1,3)) (#) (pdiff1 (f2,3)) is_partial_differentiable_in u0,3
then ( pdiff1 (f1,3) is_partial_differentiable_in u0,3 & pdiff1 (f2,3) is_partial_differentiable_in u0,3 ) by Th27;
hence (pdiff1 (f1,3)) (#) (pdiff1 (f2,3)) is_partial_differentiable_in u0,3 by PDIFF_4:30; ::_thesis: verum
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
proof
let f be PartFunc of (REAL 3),REAL; ::_thesis: 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
let u0 be Element of REAL 3; ::_thesis: ( f is_hpartial_differentiable`11_in u0 implies SVF1 (1,(pdiff1 (f,1)),u0) is_continuous_in (proj (1,3)) . u0 )
assume f is_hpartial_differentiable`11_in u0 ; ::_thesis: SVF1 (1,(pdiff1 (f,1)),u0) is_continuous_in (proj (1,3)) . u0
then pdiff1 (f,1) is_partial_differentiable_in u0,1 by Th19;
hence SVF1 (1,(pdiff1 (f,1)),u0) is_continuous_in (proj (1,3)) . u0 by PDIFF_4:31; ::_thesis: verum
end;
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
proof
let f be PartFunc of (REAL 3),REAL; ::_thesis: 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
let u0 be Element of REAL 3; ::_thesis: ( f is_hpartial_differentiable`12_in u0 implies SVF1 (2,(pdiff1 (f,1)),u0) is_continuous_in (proj (2,3)) . u0 )
assume f is_hpartial_differentiable`12_in u0 ; ::_thesis: SVF1 (2,(pdiff1 (f,1)),u0) is_continuous_in (proj (2,3)) . u0
then pdiff1 (f,1) is_partial_differentiable_in u0,2 by Th20;
hence SVF1 (2,(pdiff1 (f,1)),u0) is_continuous_in (proj (2,3)) . u0 by PDIFF_4:32; ::_thesis: verum
end;
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
proof
let f be PartFunc of (REAL 3),REAL; ::_thesis: 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
let u0 be Element of REAL 3; ::_thesis: ( f is_hpartial_differentiable`13_in u0 implies SVF1 (3,(pdiff1 (f,1)),u0) is_continuous_in (proj (3,3)) . u0 )
assume f is_hpartial_differentiable`13_in u0 ; ::_thesis: SVF1 (3,(pdiff1 (f,1)),u0) is_continuous_in (proj (3,3)) . u0
then pdiff1 (f,1) is_partial_differentiable_in u0,3 by Th21;
hence SVF1 (3,(pdiff1 (f,1)),u0) is_continuous_in (proj (3,3)) . u0 by PDIFF_4:33; ::_thesis: verum
end;
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
proof
let f be PartFunc of (REAL 3),REAL; ::_thesis: 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
let u0 be Element of REAL 3; ::_thesis: ( f is_hpartial_differentiable`21_in u0 implies SVF1 (1,(pdiff1 (f,2)),u0) is_continuous_in (proj (1,3)) . u0 )
assume f is_hpartial_differentiable`21_in u0 ; ::_thesis: SVF1 (1,(pdiff1 (f,2)),u0) is_continuous_in (proj (1,3)) . u0
then pdiff1 (f,2) is_partial_differentiable_in u0,1 by Th22;
hence SVF1 (1,(pdiff1 (f,2)),u0) is_continuous_in (proj (1,3)) . u0 by PDIFF_4:31; ::_thesis: verum
end;
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
proof
let f be PartFunc of (REAL 3),REAL; ::_thesis: 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
let u0 be Element of REAL 3; ::_thesis: ( f is_hpartial_differentiable`22_in u0 implies SVF1 (2,(pdiff1 (f,2)),u0) is_continuous_in (proj (2,3)) . u0 )
assume f is_hpartial_differentiable`22_in u0 ; ::_thesis: SVF1 (2,(pdiff1 (f,2)),u0) is_continuous_in (proj (2,3)) . u0
then pdiff1 (f,2) is_partial_differentiable_in u0,2 by Th23;
hence SVF1 (2,(pdiff1 (f,2)),u0) is_continuous_in (proj (2,3)) . u0 by PDIFF_4:32; ::_thesis: verum
end;
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
proof
let f be PartFunc of (REAL 3),REAL; ::_thesis: 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
let u0 be Element of REAL 3; ::_thesis: ( f is_hpartial_differentiable`23_in u0 implies SVF1 (3,(pdiff1 (f,2)),u0) is_continuous_in (proj (3,3)) . u0 )
assume f is_hpartial_differentiable`23_in u0 ; ::_thesis: SVF1 (3,(pdiff1 (f,2)),u0) is_continuous_in (proj (3,3)) . u0
then pdiff1 (f,2) is_partial_differentiable_in u0,3 by Th24;
hence SVF1 (3,(pdiff1 (f,2)),u0) is_continuous_in (proj (3,3)) . u0 by PDIFF_4:33; ::_thesis: verum
end;
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
proof
let f be PartFunc of (REAL 3),REAL; ::_thesis: 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
let u0 be Element of REAL 3; ::_thesis: ( f is_hpartial_differentiable`31_in u0 implies SVF1 (1,(pdiff1 (f,3)),u0) is_continuous_in (proj (1,3)) . u0 )
assume f is_hpartial_differentiable`31_in u0 ; ::_thesis: SVF1 (1,(pdiff1 (f,3)),u0) is_continuous_in (proj (1,3)) . u0
then pdiff1 (f,3) is_partial_differentiable_in u0,1 by Th25;
hence SVF1 (1,(pdiff1 (f,3)),u0) is_continuous_in (proj (1,3)) . u0 by PDIFF_4:31; ::_thesis: verum
end;
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
proof
let f be PartFunc of (REAL 3),REAL; ::_thesis: 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
let u0 be Element of REAL 3; ::_thesis: ( f is_hpartial_differentiable`32_in u0 implies SVF1 (2,(pdiff1 (f,3)),u0) is_continuous_in (proj (2,3)) . u0 )
assume f is_hpartial_differentiable`32_in u0 ; ::_thesis: SVF1 (2,(pdiff1 (f,3)),u0) is_continuous_in (proj (2,3)) . u0
then pdiff1 (f,3) is_partial_differentiable_in u0,2 by Th26;
hence SVF1 (2,(pdiff1 (f,3)),u0) is_continuous_in (proj (2,3)) . u0 by PDIFF_4:32; ::_thesis: verum
end;
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
proof
let f be PartFunc of (REAL 3),REAL; ::_thesis: 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
let u0 be Element of REAL 3; ::_thesis: ( f is_hpartial_differentiable`33_in u0 implies SVF1 (3,(pdiff1 (f,3)),u0) is_continuous_in (proj (3,3)) . u0 )
assume f is_hpartial_differentiable`33_in u0 ; ::_thesis: SVF1 (3,(pdiff1 (f,3)),u0) is_continuous_in (proj (3,3)) . u0
then pdiff1 (f,3) is_partial_differentiable_in u0,3 by Th27;
hence SVF1 (3,(pdiff1 (f,3)),u0) is_continuous_in (proj (3,3)) . u0 by PDIFF_4:33; ::_thesis: verum
end;