:: 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;