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_partial_differentiable_in u,1 holds
ex N being Neighbourhood of x0 st
( N c= dom (SVF1 (1,f,u)) & ex L being LinearFunc ex R being RestFunc st
for x being Real st x in N holds
((SVF1 (1,f,u)) . x) - ((SVF1 (1,f,u)) . x0) = (L . (x - x0)) + (R . (x - x0)) )
let u be Element of REAL 3; :: thesis: for f being PartFunc of (REAL 3),REAL st u = <*x0,y0,z0*> & f is_partial_differentiable_in u,1 holds
ex N being Neighbourhood of x0 st
( N c= dom (SVF1 (1,f,u)) & ex L being LinearFunc ex R being RestFunc st
for x being Real st x in N holds
((SVF1 (1,f,u)) . x) - ((SVF1 (1,f,u)) . x0) = (L . (x - x0)) + (R . (x - x0)) )
let f be PartFunc of (REAL 3),REAL; :: thesis: ( u = <*x0,y0,z0*> & f is_partial_differentiable_in u,1 implies ex N being Neighbourhood of x0 st
( N c= dom (SVF1 (1,f,u)) & ex L being LinearFunc ex R being RestFunc st
for x being Real st x in N holds
((SVF1 (1,f,u)) . x) - ((SVF1 (1,f,u)) . x0) = (L . (x - x0)) + (R . (x - x0)) ) )
assume that
A1: u = <*x0,y0,z0*> and
A2: f is_partial_differentiable_in u,1 ; :: thesis: ex N being Neighbourhood of x0 st
( N c= dom (SVF1 (1,f,u)) & ex L being LinearFunc ex R being RestFunc st
for x being Real st x in N holds
((SVF1 (1,f,u)) . x) - ((SVF1 (1,f,u)) . x0) = (L . (x - x0)) + (R . (x - x0)) )
consider x1, y1, z1 being Real such that
A3: ( u = <*x1,y1,z1*> & SVF1 (1,f,u) is_differentiable_in x1 ) by A2, Th7;
SVF1 (1,f,u) is_differentiable_in x0 by A1, A3, FINSEQ_1:78;
hence ex N being Neighbourhood of x0 st
( N c= dom (SVF1 (1,f,u)) & ex L being LinearFunc ex R being RestFunc st
for x being Real st x in N holds
((SVF1 (1,f,u)) . x) - ((SVF1 (1,f,u)) . x0) = (L . (x - x0)) + (R . (x - x0)) ) by FDIFF_1:def 4; :: thesis: verum