 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 G be RealNormSpace-Sequence,
     F be RealNormSpace,
     i be Element of dom G,
     x be Point of product G,
     xi be Point of G.i,
     f be PartFunc of product G,F,
     g be PartFunc of G.i,F
      st proj(i).x=xi & g=f*reproj(i,x)
 holds
    diff(g,xi) = partdiff(f,x,i)
proof
   let G be RealNormSpace-Sequence,
       F be RealNormSpace,
       i be Element of dom G,
       x be Point of product G,
       xi be Point of G.i,
       f be PartFunc of product G,F,
       g be PartFunc of G.i,F;
   i=In(i,dom G) by SUBSET_1:def 8;
   hence thesis;
end;
