let f be PartFunc of (REAL 2),REAL; :: thesis: for z0 being Element of REAL 2 st f is_hpartial_differentiable`12_in z0 holds
SVF1 (2,(pdiff1 (f,1)),z0) is_continuous_in (proj (2,2)) . z0
let z0 be Element of REAL 2; :: thesis: ( f is_hpartial_differentiable`12_in z0 implies SVF1 (2,(pdiff1 (f,1)),z0) is_continuous_in (proj (2,2)) . z0 )
assume f is_hpartial_differentiable`12_in z0 ; :: thesis: SVF1 (2,(pdiff1 (f,1)),z0) is_continuous_in (proj (2,2)) . z0
then pdiff1 (f,1) is_partial_differentiable_in z0,2 by Th10;
hence SVF1 (2,(pdiff1 (f,1)),z0) is_continuous_in (proj (2,2)) . z0 by PDIFF_2:22; :: thesis: verum