let f be PartFunc of (REAL 3),REAL; :: thesis: for u being Element of REAL 3 holds
( ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & SVF1 (3,f,u) is_differentiable_in z0 ) iff f is_partial_differentiable_in u,3 )
let u be Element of REAL 3; :: thesis: ( ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & SVF1 (3,f,u) is_differentiable_in z0 ) iff f is_partial_differentiable_in u,3 )

hereby :: thesis: ( f is_partial_differentiable_in u,3 implies ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & SVF1 (3,f,u) is_differentiable_in z0 ) )

given x0, y0, z0 being Real such that A1: ( u = <*x0,y0,z0*> & SVF1 (3,f,u) is_differentiable_in z0 ) ; :: thesis: f is_partial_differentiable_in u,3
(proj (3,3)) . u = z0 by A1, Th3;
hence f is_partial_differentiable_in u,3 by A1, PDIFF_1:def 11; :: thesis: verum

end;

assume A2: f is_partial_differentiable_in u,3 ; :: thesis: ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & SVF1 (3,f,u) is_differentiable_in z0 )
consider x0, y0, z0 being Real such that
A3: u = <*x0,y0,z0*> by FINSEQ_2:103;
(proj (3,3)) . u = z0 by A3, Th3;
then SVF1 (3,f,u) is_differentiable_in z0 by A2, PDIFF_1:def 11;
hence ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & SVF1 (3,f,u) is_differentiable_in z0 ) by A3; :: thesis: verum