let z0 be Element of REAL 2; :: thesis: for f1, f2 being PartFunc of (REAL 2),REAL st f1 is_hpartial_differentiable`21_in z0 & f2 is_hpartial_differentiable`21_in z0 holds
( (pdiff1 (f1,2)) - (pdiff1 (f2,2)) is_partial_differentiable_in z0,1 & partdiff (((pdiff1 (f1,2)) - (pdiff1 (f2,2))),z0,1) = (hpartdiff21 (f1,z0)) - (hpartdiff21 (f2,z0)) )
let f1, f2 be PartFunc of (REAL 2),REAL; :: thesis: ( f1 is_hpartial_differentiable`21_in z0 & f2 is_hpartial_differentiable`21_in z0 implies ( (pdiff1 (f1,2)) - (pdiff1 (f2,2)) is_partial_differentiable_in z0,1 & partdiff (((pdiff1 (f1,2)) - (pdiff1 (f2,2))),z0,1) = (hpartdiff21 (f1,z0)) - (hpartdiff21 (f2,z0)) ) )
assume that
A1: f1 is_hpartial_differentiable`21_in z0 and
A2: f2 is_hpartial_differentiable`21_in z0 ; :: thesis: ( (pdiff1 (f1,2)) - (pdiff1 (f2,2)) is_partial_differentiable_in z0,1 & partdiff (((pdiff1 (f1,2)) - (pdiff1 (f2,2))),z0,1) = (hpartdiff21 (f1,z0)) - (hpartdiff21 (f2,z0)) )
A3: ( pdiff1 (f1,2) is_partial_differentiable_in z0,1 & pdiff1 (f2,2) is_partial_differentiable_in z0,1 ) by A1, A2, Th11;
then partdiff (((pdiff1 (f1,2)) - (pdiff1 (f2,2))),z0,1) = (partdiff ((pdiff1 (f1,2)),z0,1)) - (partdiff ((pdiff1 (f2,2)),z0,1)) by PDIFF_1:31;
then partdiff (((pdiff1 (f1,2)) - (pdiff1 (f2,2))),z0,1) = (hpartdiff21 (f1,z0)) - (partdiff ((pdiff1 (f2,2)),z0,1)) by A1, Th15
.= (hpartdiff21 (f1,z0)) - (hpartdiff21 (f2,z0)) by A2, Th15 ;
hence ( (pdiff1 (f1,2)) - (pdiff1 (f2,2)) is_partial_differentiable_in z0,1 & partdiff (((pdiff1 (f1,2)) - (pdiff1 (f2,2))),z0,1) = (hpartdiff21 (f1,z0)) - (hpartdiff21 (f2,z0)) ) by A3, PDIFF_1:31; :: thesis: verum