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 Th12:
for i be Nat holds diff(f,Z).i is PartFunc of S,diff_SP(i,S,T)
proof
   defpred P[Nat] means
     diff(f,Z).$1 is PartFunc of S,diff_SP($1,S,T);
   diff(f,Z).0 = f|Z by Def5; then
A1:P[0] by Th7;
A2:now let i be Nat;
    assume P[i];
A3: diff(f,Z).(i+1) = (modetrans(diff(f,Z).i,S,diff_SP(i,S,T))) `| Z by Def5;
    diff_SP(i+1,S,T)
      = R_NormSpace_of_BoundedLinearOperators(S,modetrans(diff_SP(S,T).i))
          by Def2
     .= R_NormSpace_of_BoundedLinearOperators(S,diff_SP(i,S,T)) by Def1;
    hence P[i+1] by A3;
   end;
   for n be Nat holds P[n] from NAT_1:sch 2(A1,A2);
   hence thesis;
end;
