let x0, y0, z0, r be Real; :: thesis: 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; :: thesis: 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; :: thesis: ( 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 ; :: thesis: ( 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 :: thesis: ( 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) ; :: thesis: 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; :: thesis: 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)) ) ) ) ) ; :: thesis: 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; :: thesis: verum