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,2 holds

ex N being Neighbourhood of y0 st

( N c= dom (SVF1 (2,f,z)) & ex L being LinearFunc ex R being RestFunc st

for y being Real st y in N holds

((SVF1 (2,f,z)) . y) - ((SVF1 (2,f,z)) . y0) = (L . (y - y0)) + (R . (y - y0)) )

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,2 holds

ex N being Neighbourhood of y0 st

( N c= dom (SVF1 (2,f,z)) & ex L being LinearFunc ex R being RestFunc st

for y being Real st y in N holds

((SVF1 (2,f,z)) . y) - ((SVF1 (2,f,z)) . y0) = (L . (y - y0)) + (R . (y - y0)) )

let f be PartFunc of (REAL 2),REAL; :: thesis: ( z = <*x0,y0*> & f is_partial_differentiable_in z,2 implies ex N being Neighbourhood of y0 st

( N c= dom (SVF1 (2,f,z)) & ex L being LinearFunc ex R being RestFunc st

for y being Real st y in N holds

((SVF1 (2,f,z)) . y) - ((SVF1 (2,f,z)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) )

assume that

A1: z = <*x0,y0*> and

A2: f is_partial_differentiable_in z,2 ; :: thesis: ex N being Neighbourhood of y0 st

( N c= dom (SVF1 (2,f,z)) & ex L being LinearFunc ex R being RestFunc st

for y being Real st y in N holds

((SVF1 (2,f,z)) . y) - ((SVF1 (2,f,z)) . y0) = (L . (y - y0)) + (R . (y - y0)) )

ex x1, y1 being Real st

( z = <*x1,y1*> & SVF1 (2,f,z) is_differentiable_in y1 ) by A2, Th6;

then SVF1 (2,f,z) is_differentiable_in y0 by A1, FINSEQ_1:77;

hence ex N being Neighbourhood of y0 st

( N c= dom (SVF1 (2,f,z)) & ex L being LinearFunc ex R being RestFunc st

for y being Real st y in N holds

((SVF1 (2,f,z)) . y) - ((SVF1 (2,f,z)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) by FDIFF_1:def 4; :: thesis: verum

for f being PartFunc of (REAL 2),REAL st z = <*x0,y0*> & f is_partial_differentiable_in z,2 holds

ex N being Neighbourhood of y0 st

( N c= dom (SVF1 (2,f,z)) & ex L being LinearFunc ex R being RestFunc st

for y being Real st y in N holds

((SVF1 (2,f,z)) . y) - ((SVF1 (2,f,z)) . y0) = (L . (y - y0)) + (R . (y - y0)) )

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,2 holds

ex N being Neighbourhood of y0 st

( N c= dom (SVF1 (2,f,z)) & ex L being LinearFunc ex R being RestFunc st

for y being Real st y in N holds

((SVF1 (2,f,z)) . y) - ((SVF1 (2,f,z)) . y0) = (L . (y - y0)) + (R . (y - y0)) )

let f be PartFunc of (REAL 2),REAL; :: thesis: ( z = <*x0,y0*> & f is_partial_differentiable_in z,2 implies ex N being Neighbourhood of y0 st

( N c= dom (SVF1 (2,f,z)) & ex L being LinearFunc ex R being RestFunc st

for y being Real st y in N holds

((SVF1 (2,f,z)) . y) - ((SVF1 (2,f,z)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) )

assume that

A1: z = <*x0,y0*> and

A2: f is_partial_differentiable_in z,2 ; :: thesis: ex N being Neighbourhood of y0 st

( N c= dom (SVF1 (2,f,z)) & ex L being LinearFunc ex R being RestFunc st

for y being Real st y in N holds

((SVF1 (2,f,z)) . y) - ((SVF1 (2,f,z)) . y0) = (L . (y - y0)) + (R . (y - y0)) )

ex x1, y1 being Real st

( z = <*x1,y1*> & SVF1 (2,f,z) is_differentiable_in y1 ) by A2, Th6;

then SVF1 (2,f,z) is_differentiable_in y0 by A1, FINSEQ_1:77;

hence ex N being Neighbourhood of y0 st

( N c= dom (SVF1 (2,f,z)) & ex L being LinearFunc ex R being RestFunc st

for y being Real st y in N holds

((SVF1 (2,f,z)) . y) - ((SVF1 (2,f,z)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) by FDIFF_1:def 4; :: thesis: verum