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 )
thus ( ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & SVF1 (3,f,u) is_differentiable_in z0 ) implies f is_partial_differentiable_in u,3 ) by Th3; :: 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 ) )
assume A1: 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 Element of REAL such that
A2: u = <*x0,y0,z0*> by FINSEQ_2:103;
(proj (3,3)) . u = z0 by A2, Th3;
then SVF1 (3,f,u) is_differentiable_in z0 by A1;
hence ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & SVF1 (3,f,u) is_differentiable_in z0 ) by A2; :: thesis: verum