let f be PartFunc of (REAL 2),REAL; for z being Element of REAL 2 holds
( ex x0, y0 being Real st
( z = <*x0,y0*> & SVF1 (2,f,z) is_differentiable_in y0 ) iff f is_partial_differentiable_in z,2 )
let z be Element of REAL 2; ( ex x0, y0 being Real st
( z = <*x0,y0*> & SVF1 (2,f,z) is_differentiable_in y0 ) iff f is_partial_differentiable_in z,2 )
consider x0, y0 being Element of REAL such that
A1:
z = <*x0,y0*>
by FINSEQ_2:100;
thus
( ex x0, y0 being Real st
( z = <*x0,y0*> & SVF1 (2,f,z) is_differentiable_in y0 ) implies f is_partial_differentiable_in z,2 )
by Th2; ( f is_partial_differentiable_in z,2 implies ex x0, y0 being Real st
( z = <*x0,y0*> & SVF1 (2,f,z) is_differentiable_in y0 ) )
assume A2:
f is_partial_differentiable_in z,2
; ex x0, y0 being Real st
( z = <*x0,y0*> & SVF1 (2,f,z) is_differentiable_in y0 )
(proj (2,2)) . z = y0
by A1, Th2;
then
SVF1 (2,f,z) is_differentiable_in y0
by A2;
hence
ex x0, y0 being Real st
( z = <*x0,y0*> & SVF1 (2,f,z) is_differentiable_in y0 )
by A1; verum