let r be Real; :: thesis: for u0 being Element of REAL 3
for f being PartFunc of (REAL 3),REAL st f is_hpartial_differentiable`31_in u0 holds
( r (#) (pdiff1 f,3) is_partial_differentiable_in u0,1 & partdiff (r (#) (pdiff1 f,3)),u0,1 = r * (hpartdiff31 f,u0) )
let u0 be Element of REAL 3; :: thesis: for f being PartFunc of (REAL 3),REAL st f is_hpartial_differentiable`31_in u0 holds
( r (#) (pdiff1 f,3) is_partial_differentiable_in u0,1 & partdiff (r (#) (pdiff1 f,3)),u0,1 = r * (hpartdiff31 f,u0) )
let f be PartFunc of (REAL 3),REAL ; :: thesis: ( f is_hpartial_differentiable`31_in u0 implies ( r (#) (pdiff1 f,3) is_partial_differentiable_in u0,1 & partdiff (r (#) (pdiff1 f,3)),u0,1 = r * (hpartdiff31 f,u0) ) )
assume A1: f is_hpartial_differentiable`31_in u0 ; :: thesis: ( r (#) (pdiff1 f,3) is_partial_differentiable_in u0,1 & partdiff (r (#) (pdiff1 f,3)),u0,1 = r * (hpartdiff31 f,u0) )
then pdiff1 f,3 is_partial_differentiable_in u0,1 by Th11ForSecondOrder;
then ( r (#) (pdiff1 f,3) is_partial_differentiable_in u0,1 & partdiff (r (#) (pdiff1 f,3)),u0,1 = r * (partdiff (pdiff1 f,3),u0,1) ) by PDIFF_1:33;
hence ( r (#) (pdiff1 f,3) is_partial_differentiable_in u0,1 & partdiff (r (#) (pdiff1 f,3)),u0,1 = r * (hpartdiff31 f,u0) ) by A1, Th19ForSecondOrder; :: thesis: verum