reserve 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 LM120:
  for f be PartFunc of T, W,
  I be Lipschitzian LinearOperator of S, T,
  I0 be Point of R_NormSpace_of_BoundedLinearOperators(S,T)
  st I0 = I holds
  for x be Point of S
  st f is_differentiable_in I.x holds
  f*I is_differentiable_in x &
  diff(f*I,x) = diff(f,I.x)*I0
  proof
    let f be PartFunc of T, W,
    I be Lipschitzian LinearOperator of S, T,
    I0 be Point of R_NormSpace_of_BoundedLinearOperators(S,T);
    assume AS0: I0 = I;
    let x be Point of S;
    assume
    AS4: f is_differentiable_in I.x;
    X1: I is_differentiable_in x & diff(I,x) = I by LM090;
    thus f*I is_differentiable_in x by X1,AS4,NDIFF_2:13;
    thus thesis by AS0,AS4,X1,NDIFF_2:13;
  end;
