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 LM040:
  for f be PartFunc of T, W,
  I be LinearOperator of S, T
  st I is one-to-one onto isometric holds
  for x be Point of S st I.x in dom f
  holds f*I is_continuous_in x iff f is_continuous_in I.x
  proof
    let f be PartFunc of T, W,
    I be LinearOperator of S, T;
    assume that
    AS2: I is one-to-one onto and
    AS3: I is isometric;
    set g = f*I;
    let x be Point of S;
    assume
    AS6: I.x in dom f;
    P0: dom I = the carrier of S by FUNCT_2:def 1;
    hereby
      assume P1: g is_continuous_in x;
      consider J be LinearOperator of T, S such that
      P2: J = I" & J is one-to-one onto isometric by LM020,AS2,AS3;
      Q4: J is_continuous_in I.x by LM010,P2;
      P3: J.(I.x) = x by AS2,P2,P0,FUNCT_1:34;
      Q0: dom J = the carrier of T by FUNCT_2:def 1;
      Q1: J.(I.x) in dom g by P1,P3,NFCONT_1:7;
      Q3: g*J = f*(I*I") by P2,RELAT_1:36
      .= f*id (the carrier of T) by AS2,FUNCT_2:29
      .= f by FUNCT_2:17;
      J.(I.x) = J/.(I.x) & I.x = I/.x;
      hence f is_continuous_in I.x by Q3,LM030,P3,P1,Q0,Q1,Q4;
    end;
    assume
    P2: f is_continuous_in I.x;
    I.x = I/.x;
    hence g is_continuous_in x by P0,AS3,AS6,P2,LM010,LM030;
  end;
