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 f is_partial_differentiable_in z0,1 ; :: thesis: SVF1 (1,f,z0) is_continuous_in (proj (1,2)) . z0
then consider x0, y0 being Real such that
A1: z0 = <*x0,y0*> and
A2: SVF1 (1,f,z0) is_differentiable_in x0 by Th5;
SVF1 (1,f,z0) is_continuous_in x0 by A2, FDIFF_1:24;
hence SVF1 (1,f,z0) is_continuous_in (proj (1,2)) . z0 by A1, Th1; :: thesis: verum