 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
  & f is_partial_differentiable_in x,i
 holds r(#)f is_partial_differentiable_in x,i & partdiff((r(#)f),x,i)
          = r*partdiff(f,x,i)
proof
   let i0 be set;
   assume A1: i0 in dom G;
   set i=In(i0,dom G);
   assume A2: f is_partial_differentiable_in x,i0;
   r(#)(f*reproj(i,x)) = (r(#)f)*reproj(i,x) by A1,Th27;
   hence r(#)f is_partial_differentiable_in x,i0 by A2,NDIFF_1:37;
   thus partdiff(r(#)f,x,i0) = diff(r(#)(f*reproj(i,x)),proj(i).x) by A1,Th27
    .= r*partdiff(f,x,i0) by A2,NDIFF_1:37;
end;
