let x0, y0 be Real; 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 LinearFunc ex R being RestFunc 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; 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 LinearFunc ex R being RestFunc 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; ( 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 LinearFunc ex R being RestFunc 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
; ex N being Neighbourhood of x0 st
( N c= dom (SVF1 (1,f,z)) & ex L being LinearFunc ex R being RestFunc 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)) )
ex x1, y1 being Real st
( z = <*x1,y1*> & SVF1 (1,f,z) is_differentiable_in x1 )
by A2, Th5;
then
SVF1 (1,f,z) is_differentiable_in x0
by A1, FINSEQ_1:77;
hence
ex N being Neighbourhood of x0 st
( N c= dom (SVF1 (1,f,z)) & ex L being LinearFunc ex R being RestFunc 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 4; verum