let x0, y0, z0, r be Real; for u being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st u = <*x0,y0,z0*> & f is_partial_differentiable_in u,3 holds
( r = diff ((SVF1 (3,f,u)),z0) iff ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of z0 st
( N c= dom (SVF1 (3,f,u)) & ex L being LinearFunc ex R being RestFunc st
( r = L . 1 & ( for z being Real st z in N holds
((SVF1 (3,f,u)) . z) - ((SVF1 (3,f,u)) . z0) = (L . (z - z0)) + (R . (z - z0)) ) ) ) ) )
let u be Element of REAL 3; for f being PartFunc of (REAL 3),REAL st u = <*x0,y0,z0*> & f is_partial_differentiable_in u,3 holds
( r = diff ((SVF1 (3,f,u)),z0) iff ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of z0 st
( N c= dom (SVF1 (3,f,u)) & ex L being LinearFunc ex R being RestFunc st
( r = L . 1 & ( for z being Real st z in N holds
((SVF1 (3,f,u)) . z) - ((SVF1 (3,f,u)) . z0) = (L . (z - z0)) + (R . (z - z0)) ) ) ) ) )
let f be PartFunc of (REAL 3),REAL; ( u = <*x0,y0,z0*> & f is_partial_differentiable_in u,3 implies ( r = diff ((SVF1 (3,f,u)),z0) iff ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of z0 st
( N c= dom (SVF1 (3,f,u)) & ex L being LinearFunc ex R being RestFunc st
( r = L . 1 & ( for z being Real st z in N holds
((SVF1 (3,f,u)) . z) - ((SVF1 (3,f,u)) . z0) = (L . (z - z0)) + (R . (z - z0)) ) ) ) ) ) )
set F1 = SVF1 (3,f,u);
assume that
A1:
u = <*x0,y0,z0*>
and
A2:
f is_partial_differentiable_in u,3
; ( r = diff ((SVF1 (3,f,u)),z0) iff ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of z0 st
( N c= dom (SVF1 (3,f,u)) & ex L being LinearFunc ex R being RestFunc st
( r = L . 1 & ( for z being Real st z in N holds
((SVF1 (3,f,u)) . z) - ((SVF1 (3,f,u)) . z0) = (L . (z - z0)) + (R . (z - z0)) ) ) ) ) )
hereby ( ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of z0 st
( N c= dom (SVF1 (3,f,u)) & ex L being LinearFunc ex R being RestFunc st
( r = L . 1 & ( for z being Real st z in N holds
((SVF1 (3,f,u)) . z) - ((SVF1 (3,f,u)) . z0) = (L . (z - z0)) + (R . (z - z0)) ) ) ) ) implies r = diff ((SVF1 (3,f,u)),z0) )
assume A3:
r = diff (
(SVF1 (3,f,u)),
z0)
;
ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of z0 st
( N c= dom (SVF1 (3,f,u)) & ex L being LinearFunc ex R being RestFunc st
( r = L . 1 & ( for z being Real st z in N holds
((SVF1 (3,f,u)) . z) - ((SVF1 (3,f,u)) . z0) = (L . (z - z0)) + (R . (z - z0)) ) ) ) )
SVF1 (3,
f,
u)
is_differentiable_in z0
by A1, A2, Th6;
then consider N being
Neighbourhood of
z0 such that A4:
(
N c= dom (SVF1 (3,f,u)) & ex
L being
LinearFunc ex
R being
RestFunc st
(
r = L . 1 & ( for
z being
Real st
z in N holds
((SVF1 (3,f,u)) . z) - ((SVF1 (3,f,u)) . z0) = (L . (z - z0)) + (R . (z - z0)) ) ) )
by A3, FDIFF_1:def 5;
thus
ex
x0,
y0,
z0 being
Real st
(
u = <*x0,y0,z0*> & ex
N being
Neighbourhood of
z0 st
(
N c= dom (SVF1 (3,f,u)) & ex
L being
LinearFunc ex
R being
RestFunc st
(
r = L . 1 & ( for
z being
Real st
z in N holds
((SVF1 (3,f,u)) . z) - ((SVF1 (3,f,u)) . z0) = (L . (z - z0)) + (R . (z - z0)) ) ) ) )
by A1, A4;
verum
end;
given x1, y1, z1 being Real such that A5:
( u = <*x1,y1,z1*> & ex N being Neighbourhood of z1 st
( N c= dom (SVF1 (3,f,u)) & ex L being LinearFunc ex R being RestFunc st
( r = L . 1 & ( for z being Real st z in N holds
((SVF1 (3,f,u)) . z) - ((SVF1 (3,f,u)) . z1) = (L . (z - z1)) + (R . (z - z1)) ) ) ) )
; r = diff ((SVF1 (3,f,u)),z0)
A6:
z1 = z0
by A5, A1, FINSEQ_1:78;
SVF1 (3,f,u) is_differentiable_in z0
by A1, A2, Th6;
hence
r = diff ((SVF1 (3,f,u)),z0)
by A6, A5, FDIFF_1:def 5; verum