reserve E, F, G,S,T,W,Y for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve Z for Subset of S;
reserve i,n for Nat;

theorem Th22:
  for E,F,G be RealNormSpace,
      L be Lipschitzian LinearOperator of F,G
  holds
    LTRN(0,L,E) = L
  & for i be Nat
    holds
      for V be Lipschitzian LinearOperator of E,diff_SP(i,E,F)
      holds LTRN(i+1,L,E).V = LTRN(i,L,E) * V
proof
  let E,F,G be RealNormSpace,
      L be Lipschitzian LinearOperator of F,G;
  thus LTRN(0,L,E) = L by Def1;

  let i be Nat;
  consider
  K be Lipschitzian LinearOperator of diff_SP(i+1,E,F),diff_SP(i+1,E,G),
  M be Lipschitzian LinearOperator of diff_SP(i,E,F),diff_SP(i,E,G)
  such that
  A1: LTRN(L,E).(i+1) = K
    & In(LTRN(L,E).i,
          R_NormSpace_of_BoundedLinearOperators(diff_SP(i,E,F),
                                                diff_SP(i,E,G))) = M
    & for V be Lipschitzian LinearOperator of E,diff_SP(i,E,F)
      holds K.V = M*V by Def1;

  LTRN(i,L,E) in the carrier of
    R_NormSpace_of_BoundedLinearOperators(diff_SP(i,E,F),diff_SP(i,E,G))
    by LOPBAN_1:def 9;

  then
  M = LTRN(i,L,E) by A1,SUBSET_1:def 8;
  hence for V be Lipschitzian LinearOperator of E,diff_SP(i,E,F)
        holds LTRN(i+1,L,E).V = LTRN(i,L,E) * V by A1;
end;
