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 Th7:
(diff_SP(S,T)).0 = T &
(diff_SP(S,T)).1 = R_NormSpace_of_BoundedLinearOperators(S,T) &
(diff_SP(S,T)).2
   = R_NormSpace_of_BoundedLinearOperators( S,
        R_NormSpace_of_BoundedLinearOperators(S,T) )
proof
   thus A1: (diff_SP(S,T)).0 = T by Def2;
   (diff_SP(S,T)).1 = (diff_SP(S,T)).((0 qua Nat) + 1); then
   (diff_SP(S,T)).1
    = R_NormSpace_of_BoundedLinearOperators(S,modetrans(diff_SP(S,T).0))
             by Def2;
   hence A2: (diff_SP(S,T)).1
     = R_NormSpace_of_BoundedLinearOperators(S,T) by A1,Def1;
  (diff_SP(S,T)).2 = (diff_SP(S,T)).(1+1); then
  (diff_SP(S,T)).2
    = R_NormSpace_of_BoundedLinearOperators(S,modetrans(diff_SP(S,T).1))
      by Def2;
  hence (diff_SP(S,T)).2
    = R_NormSpace_of_BoundedLinearOperators(S,
          R_NormSpace_of_BoundedLinearOperators(S,T) ) by A2,Def1;
end;
