let f be PartFunc of (REAL 2),REAL ; :: thesis: for z0 being Element of REAL 2 st f is_partial_differentiable_in z0,1 holds
SVF1 1,f,z0 is_continuous_in (proj 1,2) . z0
let z0 be Element of REAL 2; :: thesis: ( f is_partial_differentiable_in z0,1 implies SVF1 1,f,z0 is_continuous_in (proj 1,2) . z0 )
assume A0: f is_partial_differentiable_in z0,1 ; :: thesis: SVF1 1,f,z0 is_continuous_in (proj 1,2) . z0
consider x0, y0 being Real such that
A1: ( z0 = <*x0,y0*> & SVF1 1,f,z0 is_differentiable_in x0 ) by A0, Def6;
SVF1 1,f,z0 is_continuous_in x0 by A1, FDIFF_1:32;
hence SVF1 1,f,z0 is_continuous_in (proj 1,2) . z0 by A1, Th1; :: thesis: verum