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

theorem Th75:
for X be Subset of REAL m, I be non empty FinSequence of NAT,
    f,g be PartFunc of REAL m,REAL st
  X is open & rng I c= Seg m
  & f is_partial_differentiable_on X,I
  & g is_partial_differentiable_on X,I
holds
    f+g is_partial_differentiable_on X,I
  & (f+g)`partial|(X,I) = f`partial|(X,I) + g`partial|(X,I)
proof
   let Z be Subset of REAL m, I be non empty FinSequence of NAT,
       f,g be PartFunc of REAL m,REAL;
   assume A1: Z is open & rng I c= Seg m
       & f is_partial_differentiable_on Z,I
       & g is_partial_differentiable_on Z,I;
   hence f+g is_partial_differentiable_on Z,I by Th74;
   reconsider k=(len I)-1 as Element of NAT by FINSEQ_1:20,INT_1:5;
A2:(PartDiffSeq(f,Z,I)).k is_partial_differentiable_on Z,I/.(k+1) &
   (PartDiffSeq(g,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;
A4:(PartDiffSeq(f+g,Z,I)).(k+1)
     = ((PartDiffSeq((f+g),Z,I)).k)`partial|(Z,I/.(k+1)) by Def7
    .= ((PartDiffSeq(f,Z,I).k)
          + (PartDiffSeq(g,Z,I).k))`partial|(Z,I/.(k+1)) by A1,Th74
    .= (PartDiffSeq(f,Z,I).k)`partial|(Z,I/.(k+1))
          + (PartDiffSeq(g,Z,I).k)`partial|(Z,I/.(k+1)) by A2,A1,A3,Th65;
   (PartDiffSeq(f,Z,I)).(k+1)
     = ((PartDiffSeq(f,Z,I)).k)`partial|(Z,I/.(k+1)) by Def7;
   hence (f+g)`partial|(Z,I) = f`partial|(Z,I) + g`partial|(Z,I) by A4,Def7;
end;
