reserve m,n for non zero Element of NAT;
reserve i,j,k for Element of NAT;
reserve Z for set;

theorem Th79:
for X be Subset of REAL m, r be Real, I be non empty FinSequence of NAT,
    f be PartFunc of REAL m,REAL st
  X is open & rng I c= Seg m & f is_partial_differentiable_on X,I
holds
   r(#)f is_partial_differentiable_on X,I
 & (r(#)f)`partial|(X,I) = r(#)(f`partial|(X,I))
proof
   let Z be Subset of REAL m, r be Real, I be non empty FinSequence of NAT,
       f be PartFunc of REAL m,REAL;
   assume A1: Z is open & rng I c= Seg m
       & f is_partial_differentiable_on Z,I;
   hence r(#)f is_partial_differentiable_on Z,I by Th78;
   reconsider k=(len I)-1 as Element of NAT by INT_1:5,FINSEQ_1:20;
A2:(PartDiffSeq(f,Z,I)).k
     is_partial_differentiable_on Z,I/.(k+1) by A1;
   1 <= k+1 by NAT_1:11; then
   I/.(k+1) in Seg m by A1,Lm6; then
A3:1<=I/.(k+1) & I/.(k+1) <= m by FINSEQ_1:1;
   (PartDiffSeq(r(#)f,Z,I)).(k+1)
     = ((PartDiffSeq((r(#)f),Z,I)).k)`partial|(Z,I/.(k+1)) by Def7
    .= (r(#)(PartDiffSeq(f,Z,I).k))`partial|(Z,I/.(k+1)) by A1,Th78
    .= r(#)((PartDiffSeq(f,Z,I).k)`partial|(Z,I/.(k+1)))
        by A2,A1,A3,Th67;
   hence (r(#)f)`partial|(Z,I) = r(#)(f`partial|(Z,I)) by Def7;
end;
