let z0 be Element of REAL 2; :: thesis: for f1, f2 being PartFunc of REAL 2, REAL st f1 is_hpartial_differentiable`11_in z0 & f2 is_hpartial_differentiable`11_in z0 holds
( (pdiff1 f1,z0) - (pdiff1 f2,z0) is_partial_differentiable`1_in z0 & partdiff1 ((pdiff1 f1,z0) - (pdiff1 f2,z0)),z0 = (hpartdiff11 f1,z0) - (hpartdiff11 f2,z0) )
let f1, f2 be PartFunc of REAL 2, REAL ; :: thesis: ( f1 is_hpartial_differentiable`11_in z0 & f2 is_hpartial_differentiable`11_in z0 implies ( (pdiff1 f1,z0) - (pdiff1 f2,z0) is_partial_differentiable`1_in z0 & partdiff1 ((pdiff1 f1,z0) - (pdiff1 f2,z0)),z0 = (hpartdiff11 f1,z0) - (hpartdiff11 f2,z0) ) )
assume that
A1: f1 is_hpartial_differentiable`11_in z0 and
A2: f2 is_hpartial_differentiable`11_in z0 ; :: thesis: ( (pdiff1 f1,z0) - (pdiff1 f2,z0) is_partial_differentiable`1_in z0 & partdiff1 ((pdiff1 f1,z0) - (pdiff1 f2,z0)),z0 = (hpartdiff11 f1,z0) - (hpartdiff11 f2,z0) )
A3: pdiff1 f1,z0 is_partial_differentiable`1_in z0 by A1, Th9;
AA: pdiff1 f2,z0 is_partial_differentiable`1_in z0 by A2, Th9;
then A4: ( (pdiff1 f1,z0) - (pdiff1 f2,z0) is_partial_differentiable`1_in z0 & partdiff1 ((pdiff1 f1,z0) - (pdiff1 f2,z0)),z0 = (partdiff1 (pdiff1 f1,z0),z0) - (partdiff1 (pdiff1 f2,z0),z0) ) by A3, PDIFF_2:19;
partdiff1 ((pdiff1 f1,z0) - (pdiff1 f2,z0)),z0 = (hpartdiff11 f1,z0) - (partdiff1 (pdiff1 f2,z0),z0) by A1, A4, Th17
.= (hpartdiff11 f1,z0) - (hpartdiff11 f2,z0) by A2, Th17 ;
hence ( (pdiff1 f1,z0) - (pdiff1 f2,z0) is_partial_differentiable`1_in z0 & partdiff1 ((pdiff1 f1,z0) - (pdiff1 f2,z0)),z0 = (hpartdiff11 f1,z0) - (hpartdiff11 f2,z0) ) by A3, AA, PDIFF_2:19; :: thesis: verum