let f be PartFunc of (REAL 3),REAL; :: thesis: for u0 being Element of REAL 3 st f is_partial_differentiable_in u0,3 holds
SVF1 (3,f,u0) is_continuous_in (proj (3,3)) . u0
let u0 be Element of REAL 3; :: thesis: ( f is_partial_differentiable_in u0,3 implies SVF1 (3,f,u0) is_continuous_in (proj (3,3)) . u0 )
assume f is_partial_differentiable_in u0,3 ; :: thesis: SVF1 (3,f,u0) is_continuous_in (proj (3,3)) . u0
then consider x0, y0, z0 being Real such that
A1: ( u0 = <*x0,y0,z0*> & SVF1 (3,f,u0) is_differentiable_in z0 ) by Th9;
SVF1 (3,f,u0) is_continuous_in z0 by A1, FDIFF_1:24;
hence SVF1 (3,f,u0) is_continuous_in (proj (3,3)) . u0 by A1, Th3; :: thesis: verum