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 LinearFunc ex R being RestFunc 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 LinearFunc ex R being RestFunc 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 LinearFunc ex R being RestFunc 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 A1: ( 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 LinearFunc ex R being RestFunc 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 LinearFunc ex R being RestFunc 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 LinearFunc ex R being RestFunc 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, A1;
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 LinearFunc ex R being RestFunc 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 Lm1, A1; :: thesis: verum

end;

given x1, y1, z1 being Real such that A2: ( u = <*x1,y1,z1*> & ex N being Neighbourhood of x1 st
( N c= dom (SVF1 (1,f,u)) & ex L being LinearFunc ex R being RestFunc 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 A2, A1, Lm1;
hence r = partdiff (f,u,1) by Th1, A1; :: thesis: verum