let f be PartFunc of (REAL 3),REAL; 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; ( 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; ( 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
; 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; verum