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,1 holds
( r = partdiff f,u,1 iff ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of x0 st
( N c= dom (SVF1 1,f,u) & ex L being LINEAR ex R being REST st
( r = L . 1 & ( for x being Real st x in N holds
((SVF1 1,f,u) . x) - ((SVF1 1,f,u) . x0) = (L . (x - x0)) + (R . (x - x0)) ) ) ) ) )
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,1 holds
( r = partdiff f,u,1 iff ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of x0 st
( N c= dom (SVF1 1,f,u) & ex L being LINEAR ex R being REST st
( r = L . 1 & ( for x being Real st x in N holds
((SVF1 1,f,u) . x) - ((SVF1 1,f,u) . x0) = (L . (x - x0)) + (R . (x - x0)) ) ) ) ) )
let f be PartFunc of (REAL 3),REAL ; :: thesis: ( u = <*x0,y0,z0*> & f is_partial_differentiable_in u,1 implies ( r = partdiff f,u,1 iff ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of x0 st
( N c= dom (SVF1 1,f,u) & ex L being LINEAR ex R being REST st
( r = L . 1 & ( for x being Real st x in N holds
((SVF1 1,f,u) . x) - ((SVF1 1,f,u) . x0) = (L . (x - x0)) + (R . (x - x0)) ) ) ) ) ) )
assume AA: ( u = <*x0,y0,z0*> & f is_partial_differentiable_in u,1 ) ; :: thesis: ( r = partdiff f,u,1 iff ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of x0 st
( N c= dom (SVF1 1,f,u) & ex L being LINEAR ex R being REST st
( r = L . 1 & ( for x being Real st x in N holds
((SVF1 1,f,u) . x) - ((SVF1 1,f,u) . x0) = (L . (x - x0)) + (R . (x - x0)) ) ) ) ) )

hereby :: thesis: ( ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of x0 st
( N c= dom (SVF1 1,f,u) & ex L being LINEAR ex R being REST st
( r = L . 1 & ( for x being Real st x in N holds
((SVF1 1,f,u) . x) - ((SVF1 1,f,u) . x0) = (L . (x - x0)) + (R . (x - x0)) ) ) ) ) implies r = partdiff f,u,1 )

assume r = partdiff f,u,1 ; :: thesis: ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of x0 st
( N c= dom (SVF1 1,f,u) & ex L being LINEAR ex R being REST st
( r = L . 1 & ( for x being Real st x in N holds
((SVF1 1,f,u) . x) - ((SVF1 1,f,u) . x0) = (L . (x - x0)) + (R . (x - x0)) ) ) ) )
then r = diff (SVF1 1,f,u),x0 by Th1, AA;
hence ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of x0 st
( N c= dom (SVF1 1,f,u) & ex L being LINEAR ex R being REST st
( r = L . 1 & ( for x being Real st x in N holds
((SVF1 1,f,u) . x) - ((SVF1 1,f,u) . x0) = (L . (x - x0)) + (R . (x - x0)) ) ) ) ) by BXXLXSdef12, AA; :: thesis: verum

end;

given x1, y1, z1 being Real such that C1: ( u = <*x1,y1,z1*> & ex N being Neighbourhood of x1 st
( N c= dom (SVF1 1,f,u) & ex L being LINEAR ex R being REST st
( r = L . 1 & ( for x being Real st x in N holds
((SVF1 1,f,u) . x) - ((SVF1 1,f,u) . x1) = (L . (x - x1)) + (R . (x - x1)) ) ) ) ) ; :: thesis: r = partdiff f,u,1
r = diff (SVF1 1,f,u),x0 by C1, AA, BXXLXSdef12;
hence r = partdiff f,u,1 by Th1, AA; :: thesis: verum