let f be PartFunc of (REAL 2),REAL; :: thesis: for z being Element of REAL 2 holds

( f is_partial_differentiable_in z,1 iff ex x0, y0 being Real st

( z = <*x0,y0*> & 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; :: thesis: ( f is_partial_differentiable_in z,1 iff ex x0, y0 being Real st

( z = <*x0,y0*> & 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)) ) ) )

A5: 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)) ) ; :: thesis: f is_partial_differentiable_in z,1

SVF1 (1,f,z) is_differentiable_in x0 by A5, FDIFF_1:def 4;

hence f is_partial_differentiable_in z,1 by A4, Th5; :: thesis: verum

( f is_partial_differentiable_in z,1 iff ex x0, y0 being Real st

( z = <*x0,y0*> & 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; :: thesis: ( f is_partial_differentiable_in z,1 iff ex x0, y0 being Real st

( z = <*x0,y0*> & 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)) ) ) )

hereby :: thesis: ( ex x0, y0 being Real st

( z = <*x0,y0*> & 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)) ) ) implies f is_partial_differentiable_in z,1 )

given x0, y0 being Real such that A4:
z = <*x0,y0*>
and ( z = <*x0,y0*> & 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)) ) ) implies f is_partial_differentiable_in z,1 )

assume A1:
f is_partial_differentiable_in z,1
; :: thesis: ex x0, y0 being Real st

( z = <*x0,y0*> & 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)) ) )

thus ex x0, y0 being Real st

( z = <*x0,y0*> & 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)) ) ) :: thesis: verum

end;( z = <*x0,y0*> & 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)) ) )

thus ex x0, y0 being Real st

( z = <*x0,y0*> & 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)) ) ) :: thesis: verum

proof

consider x0, y0 being Real such that

A2: z = <*x0,y0*> and

A3: SVF1 (1,f,z) is_differentiable_in x0 by A1, Th5;

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 A3, FDIFF_1:def 4;

hence ex x0, y0 being Real st

( z = <*x0,y0*> & 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 A2; :: thesis: verum

end;A2: z = <*x0,y0*> and

A3: SVF1 (1,f,z) is_differentiable_in x0 by A1, Th5;

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 A3, FDIFF_1:def 4;

hence ex x0, y0 being Real st

( z = <*x0,y0*> & 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 A2; :: thesis: verum

A5: 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)) ) ; :: thesis: f is_partial_differentiable_in z,1

SVF1 (1,f,z) is_differentiable_in x0 by A5, FDIFF_1:def 4;

hence f is_partial_differentiable_in z,1 by A4, Th5; :: thesis: verum