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 Th30:
  for E,F be RealNormSpace,
      i be Nat
  holds
    diff_SP(i + 1,E,F)
  = diff_SP(i,E,R_NormSpace_of_BoundedLinearOperators(E,F))
proof
  let E,F be RealNormSpace;

  defpred P[Nat] means
    diff_SP($1 + 1,E,F)
  = diff_SP($1,E,R_NormSpace_of_BoundedLinearOperators(E,F));

  diff_SP(0 + 1,E,F)
   = R_NormSpace_of_BoundedLinearOperators(E,F) by NDIFF_6:7
  .= diff_SP(0,E,R_NormSpace_of_BoundedLinearOperators(E,F)) by NDIFF_6:7;
  then A1: P[0];

  A2: for i be Nat st P[i] holds P[i+1]
  proof
    let i be Nat;
    assume A3: P[i];
    thus diff_SP((i+1)+1,E,F)
     = R_NormSpace_of_BoundedLinearOperators(E,(diff_SP(i+1,E,F)))
        by NDIFF_6:10
    .= diff_SP(i+1,E,R_NormSpace_of_BoundedLinearOperators(E,F))
        by A3,NDIFF_6:10;
  end;
  for i be Nat holds P[i] from NAT_1:sch 2(A1,A2);
  hence thesis;
end;
