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 LM155:
  for f be PartFunc of T, W,
  I be LinearOperator of S, T,
  X be set st X c= the carrier of T &
  I is one-to-one onto & I is isometric holds
  f is_differentiable_on X iff f*I is_differentiable_on I"X
  proof
    let f be PartFunc of T, W,
    I be LinearOperator of S, T,
    X be set;
    assume that
    AS1: X c= the carrier of T and
    AS2: I is one-to-one onto and
    AS3: I is isometric;
    hereby
      assume P2: f is_differentiable_on X;
      P3: I"X c= I"(dom f) by P2,RELAT_1:143;
      for x be Point of S st x in I"X
      holds (f*I) | I"X is_differentiable_in x
      proof
        let x be Point of S;
        assume x in I"X; then
        x in dom I & I.x in X by FUNCT_1:def 7; then
        P6: f|X is_differentiable_in (I.x) by P2;
        (f|X)*I is_differentiable_in x by AS2,AS3,P6,LM150;
        hence (f*I) | I"X is_differentiable_in x by FX1;
      end;
      hence f*I is_differentiable_on I"X by P3,RELAT_1:147;
    end;
    assume P2: f*I is_differentiable_on I"X; then
    K1: I"X c= I"(dom f) by RELAT_1:147;
    for y be Point of T st y in X
    holds f|X is_differentiable_in y
    proof
      let y be Point of T;
      assume P4: y in X;
      consider x be Point of S such that
      P5: y= I.x by AS2,FUNCT_2:113;
      dom I = the carrier of S by FUNCT_2:def 1;
      then x in I"X by P4,P5,FUNCT_1:def 7; then
      (f*I) | I"X is_differentiable_in x by P2; then
      (f|X)*I is_differentiable_in x by FX1;
      hence f|X is_differentiable_in y by AS2,AS3,P5,LM150;
    end;
    hence thesis by AS1,AS2,K1,FUNCT_1:88;
  end;
