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,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; 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; ( 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
; ( 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 ( 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) )
assume A4:
r = diff (
(SVF1 (1,f,z)),
x0)
;
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;
verum
end;
given x1, y1 being Real such that A5:
z = <*x1,y1*>
and
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)) ) ) )
; 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; verum