let f be PartFunc of (REAL 3),REAL ; :: thesis: for u0 being Element of REAL 3 st f is_hpartial_differentiable`31_in u0 holds
SVF1 1,(pdiff1 f,3),u0 is_continuous_in (proj 1,3) . u0
let u0 be Element of REAL 3; :: thesis: ( f is_hpartial_differentiable`31_in u0 implies SVF1 1,(pdiff1 f,3),u0 is_continuous_in (proj 1,3) . u0 )
assume f is_hpartial_differentiable`31_in u0 ; :: thesis: SVF1 1,(pdiff1 f,3),u0 is_continuous_in (proj 1,3) . u0
then pdiff1 f,3 is_partial_differentiable_in u0,1 by Th11ForSecondOrder;
hence SVF1 1,(pdiff1 f,3),u0 is_continuous_in (proj 1,3) . u0 by PDIFF_4:31; :: thesis: verum