 reserve j for set;
 reserve p,r for Real;
 reserve S,T,F for RealNormSpace;
 reserve x0 for Point of S;
 reserve g for PartFunc of S,T;
 reserve c for constant sequence of S;
 reserve R for RestFunc of S,T;
 reserve G for RealNormSpace-Sequence;
 reserve i for Element of dom G;
 reserve f for PartFunc of product G,F;
 reserve x for Element of product G;
reserve G for RealNormSpace-Sequence;
reserve F for RealNormSpace;
reserve i for Element of dom G;
reserve f,f1,f2 for PartFunc of product G, F;
reserve x for Point of product G;
reserve X for set;

theorem
for i be set st i in dom G
  & f1 is_partial_differentiable_in x,i
  & f2 is_partial_differentiable_in x,i
  holds f1+f2 is_partial_differentiable_in x,i
   & partdiff(f1+f2,x,i)=partdiff(f1,x,i)+partdiff(f2,x,i)
proof
   let i0 be set;
   set i=In(i0,dom G);
   assume A1: i0 in dom G; then
A2:(f1+f2)*reproj(i,x) = f1*reproj(i,x)+f2*reproj(i,x) by Th26;
   assume
A3: f1 is_partial_differentiable_in x,i0 &
    f2 is_partial_differentiable_in x,i0;
   hence f1+f2 is_partial_differentiable_in x,i0 by A2,NDIFF_1:35;
   thus
    partdiff(f1,x,i0)+partdiff(f2,x,i0)
    = diff(f1*reproj(i,x)+f2*reproj(i,x),proj(i).x) by A3,NDIFF_1:35
    .= partdiff((f1+f2),x,i0) by A1,Th26;
end;
