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 LM045:
  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 isometric holds
  f is_continuous_on X
  iff
  f*I is_continuous_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_continuous_on X; then
      I"X c= I"(dom f) by NFCONT_1:def 7,RELAT_1:143; then
  P3: I"X c= dom (f*I) by RELAT_1:147;
      for x be Point of S st x in I"X
      holds (f*I) | I"X is_continuous_in x
      proof
        let x be Point of S;
        assume x in I"X; then
        P5: x in dom I & I.x in X by FUNCT_1:def 7; then
        P6: f|X is_continuous_in (I.x) by P2,NFCONT_1:def 7;
        X c= dom f &
        for y be Point of T st y in X holds f|X is_continuous_in y
        by P2,NFCONT_1:def 7;
        then I.x in (dom f) /\ X by P5,XBOOLE_0:def 4;
        then I.x in dom (f|X) by RELAT_1:61;
        then (f|X)*I is_continuous_in x by AS2,AS3,P6,LM040;
        hence (f*I) | I"X is_continuous_in x by FX1;
      end;
      hence (f*I) is_continuous_on I"X by P3,NFCONT_1:def 7;
    end;
    assume P2: (f*I) is_continuous_on I"X;
    then
    I"X c= dom (f*I) &
    for x be Point of S st x in I"X holds
    (f*I) | I"X is_continuous_in x by NFCONT_1:def 7;
    then K1: I"X c= I"(dom f) by RELAT_1:147;
    P3: X c= dom f by AS1,AS2,FUNCT_1:88,K1;
    for y be Point of T st y in X holds f|X is_continuous_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_continuous_in x by P2,NFCONT_1:def 7;
      then P7: (f|X)*I is_continuous_in x by FX1;
      I.x in dom (f|X) by P3,P4,P5,RELAT_1:57;
      hence f|X is_continuous_in y by AS2,AS3,P5,P7,LM040;
    end;
    hence f is_continuous_on X by AS1,AS2,K1,FUNCT_1:88,NFCONT_1:def 7;
  end;
