let x0, y0 be Real; :: thesis: for z being Element of REAL 2
for f being PartFunc of (REAL 2),REAL st z = <*x0,y0*> & f is_partial_differentiable_in z,1 holds
ex N being Neighbourhood of x0 st
( N c= dom (SVF1 1,f,z) & ex L being LINEAR ex R being REST st
for x being Real st x in N holds
((SVF1 1,f,z) . x) - ((SVF1 1,f,z) . x0) = (L . (x - x0)) + (R . (x - x0)) )
let z be Element of REAL 2; :: thesis: for f being PartFunc of (REAL 2),REAL st z = <*x0,y0*> & f is_partial_differentiable_in z,1 holds
ex N being Neighbourhood of x0 st
( N c= dom (SVF1 1,f,z) & ex L being LINEAR ex R being REST st
for x being Real st x in N holds
((SVF1 1,f,z) . x) - ((SVF1 1,f,z) . x0) = (L . (x - x0)) + (R . (x - x0)) )
let f be PartFunc of (REAL 2),REAL ; :: thesis: ( z = <*x0,y0*> & f is_partial_differentiable_in z,1 implies ex N being Neighbourhood of x0 st
( N c= dom (SVF1 1,f,z) & ex L being LINEAR ex R being REST st
for x being Real st x in N holds
((SVF1 1,f,z) . x) - ((SVF1 1,f,z) . x0) = (L . (x - x0)) + (R . (x - x0)) ) )
assume that
A1: z = <*x0,y0*> and
A2: f is_partial_differentiable_in z,1 ; :: thesis: ex N being Neighbourhood of x0 st
( N c= dom (SVF1 1,f,z) & ex L being LINEAR ex R being REST st
for x being Real st x in N holds
((SVF1 1,f,z) . x) - ((SVF1 1,f,z) . x0) = (L . (x - x0)) + (R . (x - x0)) )
consider x1, y1 being Real such that
A3: ( z = <*x1,y1*> & SVF1 1,f,z is_differentiable_in x1 ) by A2, Def6;
SVF1 1,f,z is_differentiable_in x0 by A1, A3, FINSEQ_1:98;
hence ex N being Neighbourhood of x0 st
( N c= dom (SVF1 1,f,z) & ex L being LINEAR ex R being REST st
for x being Real st x in N holds
((SVF1 1,f,z) . x) - ((SVF1 1,f,z) . x0) = (L . (x - x0)) + (R . (x - x0)) ) by FDIFF_1:def 5; :: thesis: verum