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 = partdiff (f,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

( 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 = partdiff (f,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

( 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 = partdiff (f,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

( 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)) ) ) ) ) ) )

assume that

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

A2: f is_partial_differentiable_in z,1 ; :: thesis: ( r = partdiff (f,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

( 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)) ) ) ) ) )

( 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 = partdiff (f,z,1)

r = diff ((SVF1 (1,f,z)),x0) by A1, A2, A3, Lm1;

hence r = partdiff (f,z,1) by A1, Th1; :: thesis: verum

for f being PartFunc of (REAL 2),REAL st z = <*x0,y0*> & f is_partial_differentiable_in z,1 holds

( r = partdiff (f,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

( 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 = partdiff (f,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

( 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 = partdiff (f,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

( 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)) ) ) ) ) ) )

assume that

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

A2: f is_partial_differentiable_in z,1 ; :: thesis: ( r = partdiff (f,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

( 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)) ) ) ) ) )

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 = partdiff (f,z,1) )

given x1, y1 being Real such that A3:
( z = <*x1,y1*> & ex N being Neighbourhood of x1 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 = partdiff (f,z,1) )

assume
r = partdiff (f,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

( 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)) ) ) ) )

then r = diff ((SVF1 (1,f,z)),x0) by A1, Th1;

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, A2, Lm1; :: 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)) ) ) ) )

then r = diff ((SVF1 (1,f,z)),x0) by A1, Th1;

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, A2, Lm1; :: thesis: verum

( 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 = partdiff (f,z,1)

r = diff ((SVF1 (1,f,z)),x0) by A1, A2, A3, Lm1;

hence r = partdiff (f,z,1) by A1, Th1; :: thesis: verum