let x0, y0, r 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

( r = diff ((SVF1 (2,f,z)),y0) iff ex x0, y0 being Real st

( z = <*x0,y0*> & ex N being Neighbourhood of y0 st

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

( r = L . 1 & ( 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

( r = diff ((SVF1 (2,f,z)),y0) iff ex x0, y0 being Real st

( z = <*x0,y0*> & ex N being Neighbourhood of y0 st

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

( r = L . 1 & ( 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 ( r = diff ((SVF1 (2,f,z)),y0) iff ex x0, y0 being Real st

( z = <*x0,y0*> & ex N being Neighbourhood of y0 st

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

( r = L . 1 & ( 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)) ) ) ) ) ) )

set F1 = SVF1 (2,f,z);

assume that

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

A2: f is_partial_differentiable_in z,2 ; :: thesis: ( r = diff ((SVF1 (2,f,z)),y0) iff ex x0, y0 being Real st

( z = <*x0,y0*> & ex N being Neighbourhood of y0 st

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

( r = L . 1 & ( 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)) ) ) ) ) )

A3: SVF1 (2,f,z) is_differentiable_in y0 by A1, A2, Th4;

A6: ex N being Neighbourhood of y1 st

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

( r = L . 1 & ( for y being Real st y in N holds

((SVF1 (2,f,z)) . y) - ((SVF1 (2,f,z)) . y1) = (L . (y - y1)) + (R . (y - y1)) ) ) ) ; :: thesis: r = diff ((SVF1 (2,f,z)),y0)

y1 = y0 by A1, A5, FINSEQ_1:77;

hence r = diff ((SVF1 (2,f,z)),y0) by A6, A3, FDIFF_1:def 5; :: thesis: verum

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

( r = diff ((SVF1 (2,f,z)),y0) iff ex x0, y0 being Real st

( z = <*x0,y0*> & ex N being Neighbourhood of y0 st

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

( r = L . 1 & ( 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

( r = diff ((SVF1 (2,f,z)),y0) iff ex x0, y0 being Real st

( z = <*x0,y0*> & ex N being Neighbourhood of y0 st

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

( r = L . 1 & ( 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 ( r = diff ((SVF1 (2,f,z)),y0) iff ex x0, y0 being Real st

( z = <*x0,y0*> & ex N being Neighbourhood of y0 st

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

( r = L . 1 & ( 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)) ) ) ) ) ) )

set F1 = SVF1 (2,f,z);

assume that

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

A2: f is_partial_differentiable_in z,2 ; :: thesis: ( r = diff ((SVF1 (2,f,z)),y0) iff ex x0, y0 being Real st

( z = <*x0,y0*> & ex N being Neighbourhood of y0 st

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

( r = L . 1 & ( 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)) ) ) ) ) )

A3: SVF1 (2,f,z) is_differentiable_in y0 by A1, A2, Th4;

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

( z = <*x0,y0*> & ex N being Neighbourhood of y0 st

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

( r = L . 1 & ( 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)) ) ) ) ) implies r = diff ((SVF1 (2,f,z)),y0) )

given x1, y1 being Real such that A5:
z = <*x1,y1*>
and ( z = <*x0,y0*> & ex N being Neighbourhood of y0 st

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

( r = L . 1 & ( 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)) ) ) ) ) implies r = diff ((SVF1 (2,f,z)),y0) )

assume A4:
r = diff ((SVF1 (2,f,z)),y0)
; :: thesis: ex x0, y0 being Real st

( z = <*x0,y0*> & ex N being Neighbourhood of y0 st

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

( r = L . 1 & ( 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)) ) ) ) )

SVF1 (2,f,z) is_differentiable_in y0 by A1, A2, Th4;

then ex N being Neighbourhood of y0 st

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

( r = L . 1 & ( 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 A4, FDIFF_1:def 5;

hence ex x0, y0 being Real st

( z = <*x0,y0*> & ex N being Neighbourhood of y0 st

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

( r = L . 1 & ( 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 A1; :: thesis: verum

end;( z = <*x0,y0*> & ex N being Neighbourhood of y0 st

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

( r = L . 1 & ( 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)) ) ) ) )

SVF1 (2,f,z) is_differentiable_in y0 by A1, A2, Th4;

then ex N being Neighbourhood of y0 st

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

( r = L . 1 & ( 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 A4, FDIFF_1:def 5;

hence ex x0, y0 being Real st

( z = <*x0,y0*> & ex N being Neighbourhood of y0 st

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

( r = L . 1 & ( 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 A1; :: thesis: verum

A6: ex N being Neighbourhood of y1 st

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

( r = L . 1 & ( for y being Real st y in N holds

((SVF1 (2,f,z)) . y) - ((SVF1 (2,f,z)) . y1) = (L . (y - y1)) + (R . (y - y1)) ) ) ) ; :: thesis: r = diff ((SVF1 (2,f,z)),y0)

y1 = y0 by A1, A5, FINSEQ_1:77;

hence r = diff ((SVF1 (2,f,z)),y0) by A6, A3, FDIFF_1:def 5; :: thesis: verum