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 Th9:
for i be Nat holds
 ex H be RealNormSpace st H = (diff_SP(S,T)).i
 & (diff_SP(S,T)).(i+1) = R_NormSpace_of_BoundedLinearOperators(S,H)
proof
   let i be Nat;
   take H = modetrans(diff_SP(S,T).i);
   thus H = diff_SP(S,T).i by Def1,Th8;
   thus (diff_SP(S,T)).(i+1)
    = R_NormSpace_of_BoundedLinearOperators(S,H) by Def2;
end;
