let f be PartFunc of (REAL 2),REAL ; :: thesis: for z being Element of REAL 2 holds
( ex x0, y0 being Real st
( z = <*x0,y0*> & SVF1 1,f,z is_differentiable_in x0 ) iff f is_partial_differentiable_in z,1 )
let z be Element of REAL 2; :: thesis: ( ex x0, y0 being Real st
( z = <*x0,y0*> & SVF1 1,f,z is_differentiable_in x0 ) iff f is_partial_differentiable_in z,1 )
z is Tuple of 2, REAL by FINSEQ_2:151;
then consider x0, y0 being Real such that
A1: z = <*x0,y0*> by FINSEQ_2:120;

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

given x0, y0 being Real such that A2: z = <*x0,y0*> and
A3: SVF1 1,f,z is_differentiable_in x0 ; :: thesis: f is_partial_differentiable_in z,1
(proj 1,2) . z = x0 by A2, Th1;
hence f is_partial_differentiable_in z,1 by A3, PDIFF_1:def 11; :: thesis: verum

end;

assume A4: f is_partial_differentiable_in z,1 ; :: thesis: ex x0, y0 being Real st
( z = <*x0,y0*> & SVF1 1,f,z is_differentiable_in x0 )
(proj 1,2) . z = x0 by A1, Th1;
then SVF1 1,f,z is_differentiable_in x0 by A4, PDIFF_1:def 11;
hence ex x0, y0 being Real st
( z = <*x0,y0*> & SVF1 1,f,z is_differentiable_in x0 ) by A1; :: thesis: verum