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,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 LINEAR ex R being REST 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; :: thesis: 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 LINEAR ex R being REST 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 ; :: thesis: ( 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 LINEAR ex R being REST 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 AA: ( u = <*x0,y0,z0*> & f is_partial_differentiable_in u,2 ) ; :: thesis: ( 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 LINEAR ex R being REST 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 :: thesis: ( 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 LINEAR ex R being REST 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 ; :: thesis: 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 LINEAR ex R being REST 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, AA;
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 LINEAR ex R being REST 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 BXXLXSdef13, AA; :: thesis: verum

end;

given x1, y1, z1 being Real such that C1: ( u = <*x1,y1,z1*> & ex N being Neighbourhood of y1 st
( N c= dom (SVF1 2,f,u) & ex L being LINEAR ex R being REST 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)) ) ) ) ) ; :: thesis: r = partdiff f,u,2
r = diff (SVF1 2,f,u),y0 by C1, AA, BXXLXSdef13;
hence r = partdiff f,u,2 by Th2, AA; :: thesis: verum