let f be PartFunc of (REAL 3),REAL; :: thesis: for u being Element of REAL 3 holds
( f is_partial_differentiable_in 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 LinearFunc ex R being RestFunc st
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: ( f is_partial_differentiable_in 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 LinearFunc ex R being RestFunc st
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 LinearFunc ex R being RestFunc st
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 f is_partial_differentiable_in u,2 )

assume A1: f is_partial_differentiable_in 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 LinearFunc ex R being RestFunc st
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)) ) )
thus 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 LinearFunc ex R being RestFunc st
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)) ) ) :: thesis: verum

proof

consider x0, y0, z0 being Real such that
A2: ( u = <*x0,y0,z0*> & SVF1 (2,f,u) is_differentiable_in y0 ) by A1, Th8;
ex N being Neighbourhood of y0 st
( N c= dom (SVF1 (2,f,u)) & ex L being LinearFunc ex R being RestFunc st
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 A2, FDIFF_1:def 4;
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 LinearFunc ex R being RestFunc st
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 A2; :: thesis: verum

end;

end;

assume 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 LinearFunc ex R being RestFunc st
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)) ) ) ; :: thesis: f is_partial_differentiable_in u,2
then consider x0, y0, z0 being Real such that
A3: ( u = <*x0,y0,z0*> & ex N being Neighbourhood of y0 st
( N c= dom (SVF1 (2,f,u)) & ex L being LinearFunc ex R being RestFunc st
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)) ) ) ;
consider N being Neighbourhood of y0 such that
A4: ( N c= dom (SVF1 (2,f,u)) & ex L being LinearFunc ex R being RestFunc st
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 A3;
SVF1 (2,f,u) is_differentiable_in y0 by A4, FDIFF_1:def 4;
hence f is_partial_differentiable_in u,2 by A3, Th8; :: thesis: verum