let x0, y0, r be Real; 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 = partdiff (f,z,2) 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; for f being PartFunc of (REAL 2),REAL st z = <*x0,y0*> & f is_partial_differentiable_in z,2 holds
( r = partdiff (f,z,2) 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; ( z = <*x0,y0*> & f is_partial_differentiable_in z,2 implies ( r = partdiff (f,z,2) 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)) ) ) ) ) ) )
assume that
A1:
z = <*x0,y0*>
and
A2:
f is_partial_differentiable_in z,2
; ( r = partdiff (f,z,2) 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)) ) ) ) ) )
hereby ( 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 = partdiff (f,z,2) )
assume
r = partdiff (
f,
z,2)
;
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)) ) ) ) )then
r = diff (
(SVF1 (2,f,z)),
y0)
by A1, Th2;
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, A2, Lm2;
verum
end;
given x1, y1 being Real such that A3:
( z = <*x1,y1*> & 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)) ) ) ) )
; r = partdiff (f,z,2)
r = diff ((SVF1 (2,f,z)),y0)
by A1, A2, A3, Lm2;
hence
r = partdiff (f,z,2)
by A1, Th2; verum