let x0, y0, z0 be Real; for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st u = <*x0,y0,z0*> & f is_partial_differentiable_in u,3 holds
ex N being Neighbourhood of z0 st
( N c= dom (SVF1 (3,f,u)) & ex L being LinearFunc ex R being RestFunc st
for z being Real st z in N holds
((SVF1 (3,f,u)) . z) - ((SVF1 (3,f,u)) . z0) = (L . (z - z0)) + (R . (z - z0)) )
let u be Element of REAL 3; for f being PartFunc of (REAL 3),REAL st u = <*x0,y0,z0*> & f is_partial_differentiable_in u,3 holds
ex N being Neighbourhood of z0 st
( N c= dom (SVF1 (3,f,u)) & ex L being LinearFunc ex R being RestFunc st
for z being Real st z in N holds
((SVF1 (3,f,u)) . z) - ((SVF1 (3,f,u)) . z0) = (L . (z - z0)) + (R . (z - z0)) )
let f be PartFunc of (REAL 3),REAL; ( u = <*x0,y0,z0*> & f is_partial_differentiable_in u,3 implies ex N being Neighbourhood of z0 st
( N c= dom (SVF1 (3,f,u)) & ex L being LinearFunc ex R being RestFunc st
for z being Real st z in N holds
((SVF1 (3,f,u)) . z) - ((SVF1 (3,f,u)) . z0) = (L . (z - z0)) + (R . (z - z0)) ) )
assume that
A1:
u = <*x0,y0,z0*>
and
A2:
f is_partial_differentiable_in u,3
; ex N being Neighbourhood of z0 st
( N c= dom (SVF1 (3,f,u)) & ex L being LinearFunc ex R being RestFunc st
for z being Real st z in N holds
((SVF1 (3,f,u)) . z) - ((SVF1 (3,f,u)) . z0) = (L . (z - z0)) + (R . (z - z0)) )
consider x1, y1, z1 being Real such that
A3:
( u = <*x1,y1,z1*> & SVF1 (3,f,u) is_differentiable_in z1 )
by A2, Th9;
SVF1 (3,f,u) is_differentiable_in z0
by A1, A3, FINSEQ_1:78;
hence
ex N being Neighbourhood of z0 st
( N c= dom (SVF1 (3,f,u)) & ex L being LinearFunc ex R being RestFunc st
for z being Real st z in N holds
((SVF1 (3,f,u)) . z) - ((SVF1 (3,f,u)) . z0) = (L . (z - z0)) + (R . (z - z0)) )
by FDIFF_1:def 4; verum