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 Th8:
for i be Nat holds (diff_SP(S,T)).i is RealNormSpace
proof
   defpred P[Nat] means (diff_SP(S,T)).$1 is RealNormSpace;
A1:P[0] by Th7;
A2:now let i be Nat;
    assume P[i]; then
    reconsider H = diff_SP(S,T).i as RealNormSpace;
    modetrans(diff_SP(S,T).i) = H by Def1; then
    (diff_SP(S,T)).(i+1)
      = R_NormSpace_of_BoundedLinearOperators(S,H) by Def2;
    hence P[i+1];
   end;
   for n be Nat holds P[n] from NAT_1:sch 2(A1,A2);
   hence thesis;
end;
