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,2 holds
( r = partdiff (f,u,2) iff ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of y0 st
( N c= dom (SVF1 (2,f,u)) & ex L being LinearFunc ex R being RestFunc st
( r = L . 1 & ( for y being Real st y in N holds
((SVF1 (2,f,u)) . y) - ((SVF1 (2,f,u)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) ) ) )
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,2 holds
( r = partdiff (f,u,2) iff ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of y0 st
( N c= dom (SVF1 (2,f,u)) & ex L being LinearFunc ex R being RestFunc st
( r = L . 1 & ( for y being Real st y in N holds
((SVF1 (2,f,u)) . y) - ((SVF1 (2,f,u)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) ) ) )
let f be PartFunc of (REAL 3),REAL; ( u = <*x0,y0,z0*> & f is_partial_differentiable_in u,2 implies ( r = partdiff (f,u,2) iff ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of y0 st
( N c= dom (SVF1 (2,f,u)) & ex L being LinearFunc ex R being RestFunc st
( r = L . 1 & ( for y being Real st y in N holds
((SVF1 (2,f,u)) . y) - ((SVF1 (2,f,u)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) ) ) ) )
assume A1:
( u = <*x0,y0,z0*> & f is_partial_differentiable_in u,2 )
; ( r = partdiff (f,u,2) iff ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of y0 st
( N c= dom (SVF1 (2,f,u)) & ex L being LinearFunc ex R being RestFunc st
( r = L . 1 & ( for y being Real st y in N holds
((SVF1 (2,f,u)) . y) - ((SVF1 (2,f,u)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) ) ) )
hereby ( ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of y0 st
( N c= dom (SVF1 (2,f,u)) & ex L being LinearFunc ex R being RestFunc st
( r = L . 1 & ( for y being Real st y in N holds
((SVF1 (2,f,u)) . y) - ((SVF1 (2,f,u)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) ) ) implies r = partdiff (f,u,2) )
assume
r = partdiff (
f,
u,2)
;
ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of y0 st
( N c= dom (SVF1 (2,f,u)) & ex L being LinearFunc ex R being RestFunc st
( r = L . 1 & ( for y being Real st y in N holds
((SVF1 (2,f,u)) . y) - ((SVF1 (2,f,u)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) ) )then
r = diff (
(SVF1 (2,f,u)),
y0)
by Th2, A1;
hence
ex
x0,
y0,
z0 being
Real st
(
u = <*x0,y0,z0*> & ex
N being
Neighbourhood of
y0 st
(
N c= dom (SVF1 (2,f,u)) & ex
L being
LinearFunc ex
R being
RestFunc st
(
r = L . 1 & ( for
y being
Real st
y in N holds
((SVF1 (2,f,u)) . y) - ((SVF1 (2,f,u)) . y0) = (L . (y - y0)) + (R . (y - y0)) ) ) ) )
by Lm2, A1;
verum
end;
given x1, y1, z1 being Real such that A2:
( u = <*x1,y1,z1*> & ex N being Neighbourhood of y1 st
( N c= dom (SVF1 (2,f,u)) & ex L being LinearFunc ex R being RestFunc st
( r = L . 1 & ( for y being Real st y in N holds
((SVF1 (2,f,u)) . y) - ((SVF1 (2,f,u)) . y1) = (L . (y - y1)) + (R . (y - y1)) ) ) ) )
; r = partdiff (f,u,2)
r = diff ((SVF1 (2,f,u)),y0)
by A2, A1, Lm2;
hence
r = partdiff (f,u,2)
by Th2, A1; verum