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 LINEAR ex R being REST 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 LINEAR ex R being REST 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 LINEAR ex R being REST 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 LINEAR ex R being REST 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 LINEAR ex R being REST 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 AB: 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 LINEAR ex R being REST 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, Lem3;
then consider N being Neighbourhood of z0 such that
A4: ( N c= dom (SVF1 3,f,u) & ex L being LINEAR ex R being REST 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 AB, FDIFF_1:def 6;
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 LINEAR ex R being REST 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 C1: ( u = <*x1,y1,z1*> & ex N being Neighbourhood of z1 st
( N c= dom (SVF1 3,f,u) & ex L being LINEAR ex R being REST 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
D1: z1 = z0 by C1, A1, FINSEQ_1:99;
SVF1 3,f,u is_differentiable_in z0 by A1, A2, Lem3;
hence r = diff (SVF1 3,f,u),z0 by D1, C1, FDIFF_1:def 6; :: thesis: verum