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 )
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 (1,f,z) is_differentiable_in x0 ) implies f is_partial_differentiable_in z,1 ) by Th1; :: 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 ) )
assume A2: 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 A2;
hence ex x0, y0 being Real st
( z = <*x0,y0*> & SVF1 (1,f,z) is_differentiable_in x0 ) by A1; :: thesis: verum