reserve S,T for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve Z for Subset of S;
reserve i,n for Nat;

theorem Th13:
diff(f,i+1,Z) = diff(f,i,Z) `|Z
proof
A1:diff(f,Z).i is PartFunc of S,diff_SP(i,S,T) by Th12;
A2:modetrans(diff_SP(S,T).i) = diff_SP(S,T).i by Def1,Th8; then
   modetrans(diff(f,Z).i,S,modetrans(diff_SP(S,T).i))
     = diff(f,Z).i by A1,Def4;
   hence thesis by A2,Def5;
end;
