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 1,f,u is_differentiable_in x0 ) iff f is_partial_differentiable_in u,1 )
let u be Element of REAL 3; :: thesis: ( ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & SVF1 1,f,u is_differentiable_in x0 ) iff f is_partial_differentiable_in u,1 )

hereby :: thesis: ( f is_partial_differentiable_in u,1 implies ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & SVF1 1,f,u is_differentiable_in x0 ) )

given x0, y0, z0 being Real such that A1: ( u = <*x0,y0,z0*> & SVF1 1,f,u is_differentiable_in x0 ) ; :: thesis: f is_partial_differentiable_in u,1
AA: (proj 1,3) . u = x0 by A1, Th1;
thus f is_partial_differentiable_in u,1 by A1, AA, PDIFF_1:def 11; :: thesis: verum

end;

assume A2: f is_partial_differentiable_in u,1 ; :: thesis: ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & SVF1 1,f,u is_differentiable_in x0 )
consider x0, y0, z0 being Real such that
A3: u = <*x0,y0,z0*> by FINSEQ_2:123;
A4: (proj 1,3) . u = x0 by A3, Th1;
SVF1 1,f,u is_differentiable_in x0 by A2, A4, PDIFF_1:def 11;
hence ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & SVF1 1,f,u is_differentiable_in x0 ) by A3; :: thesis: verum