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 = partdiff f,u,3 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 = partdiff f,u,3 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 = partdiff f,u,3 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)) ) ) ) ) ) )
assume AA: ( u = <*x0,y0,z0*> & f is_partial_differentiable_in u,3 ) ; :: thesis: ( r = partdiff f,u,3 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 = partdiff f,u,3 )

assume r = partdiff f,u,3 ; :: 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)) ) ) ) )
then r = diff (SVF1 3,f,u),z0 by Th3, AA;
hence 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 BXXLXSdef13ForZ, AA; :: 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 = partdiff f,u,3
r = diff (SVF1 3,f,u),z0 by C1, AA, BXXLXSdef13ForZ;
hence r = partdiff f,u,3 by Th3, AA; :: thesis: verum