let z0 be Element of REAL 2; :: thesis: for f1, f2 being PartFunc of REAL 2, REAL st f1 is_hpartial_differentiable`12_in z0 & f2 is_hpartial_differentiable`12_in z0 holds
( (pdiff1 f1,z0) - (pdiff1 f2,z0) is_partial_differentiable`2_in z0 & partdiff2 ((pdiff1 f1,z0) - (pdiff1 f2,z0)),z0 = (hpartdiff12 f1,z0) - (hpartdiff12 f2,z0) )
let f1, f2 be PartFunc of REAL 2, REAL ; :: thesis: ( f1 is_hpartial_differentiable`12_in z0 & f2 is_hpartial_differentiable`12_in z0 implies ( (pdiff1 f1,z0) - (pdiff1 f2,z0) is_partial_differentiable`2_in z0 & partdiff2 ((pdiff1 f1,z0) - (pdiff1 f2,z0)),z0 = (hpartdiff12 f1,z0) - (hpartdiff12 f2,z0) ) )
assume that
A1: f1 is_hpartial_differentiable`12_in z0 and
A2: f2 is_hpartial_differentiable`12_in z0 ; :: thesis: ( (pdiff1 f1,z0) - (pdiff1 f2,z0) is_partial_differentiable`2_in z0 & partdiff2 ((pdiff1 f1,z0) - (pdiff1 f2,z0)),z0 = (hpartdiff12 f1,z0) - (hpartdiff12 f2,z0) )
A3: pdiff1 f1,z0 is_partial_differentiable`2_in z0 by A1, Th10;
AA: pdiff1 f2,z0 is_partial_differentiable`2_in z0 by A2, Th10;
then A4: ( (pdiff1 f1,z0) - (pdiff1 f2,z0) is_partial_differentiable`2_in z0 & partdiff2 ((pdiff1 f1,z0) - (pdiff1 f2,z0)),z0 = (partdiff2 (pdiff1 f1,z0),z0) - (partdiff2 (pdiff1 f2,z0),z0) ) by A3, PDIFF_2:20;
partdiff2 ((pdiff1 f1,z0) - (pdiff1 f2,z0)),z0 = (hpartdiff12 f1,z0) - (partdiff2 (pdiff1 f2,z0),z0) by A1, A4, Th18
.= (hpartdiff12 f1,z0) - (hpartdiff12 f2,z0) by A2, Th18 ;
hence ( (pdiff1 f1,z0) - (pdiff1 f2,z0) is_partial_differentiable`2_in z0 & partdiff2 ((pdiff1 f1,z0) - (pdiff1 f2,z0)),z0 = (hpartdiff12 f1,z0) - (hpartdiff12 f2,z0) ) by A3, AA, PDIFF_2:20; :: thesis: verum