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,1 holds

( r = diff ((SVF1 (1,f,z)),x0) 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

( r = L . 1 & ( 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

( r = diff ((SVF1 (1,f,z)),x0) 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

( r = L . 1 & ( 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 ( r = diff ((SVF1 (1,f,z)),x0) 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

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

set F1 = SVF1 (1,f,z);

assume that

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

A2: f is_partial_differentiable_in z,1 ; :: thesis: ( r = diff ((SVF1 (1,f,z)),x0) 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

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

A3: SVF1 (1,f,z) is_differentiable_in x0 by A1, A2, Th3;

A6: ex N being Neighbourhood of x1 st

( N c= dom (SVF1 (1,f,z)) & ex L being LinearFunc ex R being RestFunc st

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

((SVF1 (1,f,z)) . x) - ((SVF1 (1,f,z)) . x1) = (L . (x - x1)) + (R . (x - x1)) ) ) ) ; :: thesis: r = diff ((SVF1 (1,f,z)),x0)

x1 = x0 by A1, A5, FINSEQ_1:77;

hence r = diff ((SVF1 (1,f,z)),x0) 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,1 holds

( r = diff ((SVF1 (1,f,z)),x0) 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

( r = L . 1 & ( 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

( r = diff ((SVF1 (1,f,z)),x0) 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

( r = L . 1 & ( 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 ( r = diff ((SVF1 (1,f,z)),x0) 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

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

set F1 = SVF1 (1,f,z);

assume that

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

A2: f is_partial_differentiable_in z,1 ; :: thesis: ( r = diff ((SVF1 (1,f,z)),x0) 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

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

A3: SVF1 (1,f,z) is_differentiable_in x0 by A1, A2, Th3;

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

( r = L . 1 & ( 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 r = diff ((SVF1 (1,f,z)),x0) )

given x1, y1 being Real such that A5:
z = <*x1,y1*>
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

( r = L . 1 & ( 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 r = diff ((SVF1 (1,f,z)),x0) )

assume A4:
r = diff ((SVF1 (1,f,z)),x0)
; :: 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

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

SVF1 (1,f,z) is_differentiable_in x0 by A1, A2, Th3;

then ex N being Neighbourhood of x0 st

( N c= dom (SVF1 (1,f,z)) & ex L being LinearFunc ex R being RestFunc st

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

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

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

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

SVF1 (1,f,z) is_differentiable_in x0 by A1, A2, Th3;

then ex N being Neighbourhood of x0 st

( N c= dom (SVF1 (1,f,z)) & ex L being LinearFunc ex R being RestFunc st

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

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

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

A6: ex N being Neighbourhood of x1 st

( N c= dom (SVF1 (1,f,z)) & ex L being LinearFunc ex R being RestFunc st

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

((SVF1 (1,f,z)) . x) - ((SVF1 (1,f,z)) . x1) = (L . (x - x1)) + (R . (x - x1)) ) ) ) ; :: thesis: r = diff ((SVF1 (1,f,z)),x0)

x1 = x0 by A1, A5, FINSEQ_1:77;

hence r = diff ((SVF1 (1,f,z)),x0) by A6, A3, FDIFF_1:def 5; :: thesis: verum