let f be PartFunc of (REAL 3),REAL; :: thesis: for u being Element of REAL 3 holds
( f is_partial_differentiable_in 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
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: ( f is_partial_differentiable_in 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
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
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 f is_partial_differentiable_in u,1 )

assume A1: f is_partial_differentiable_in 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
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)) ) )
thus 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
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)) ) ) :: thesis: verum

proof

consider x0, y0, z0 being Real such that
A2: ( u = <*x0,y0,z0*> & SVF1 (1,f,u) is_differentiable_in x0 ) by A1, Th7;
ex N being Neighbourhood of x0 st
( N c= dom (SVF1 (1,f,u)) & ex L being LinearFunc ex R being RestFunc st
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 A2, FDIFF_1:def 4;
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
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 A2; :: thesis: verum

end;

end;

assume 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
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)) ) ) ; :: thesis: f is_partial_differentiable_in u,1
then consider x0, y0, z0 being Real such that
A3: ( 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
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)) ) ) ;
consider N being Neighbourhood of x0 such that
A4: ( N c= dom (SVF1 (1,f,u)) & ex L being LinearFunc ex R being RestFunc st
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 A3;
SVF1 (1,f,u) is_differentiable_in x0 by A4, FDIFF_1:def 4;
hence f is_partial_differentiable_in u,1 by A3, Th7; :: thesis: verum