reserve p for Real;
reserve S,T for RealNormSpace;
reserve x0 for Point of S;
reserve f for PartFunc of S,T;
reserve c for constant sequence of S;
reserve R for RestFunc of S,T;
reserve U for RealNormSpace;

theorem
  for R be RestFunc of T,U st R/.0.T=0.U
  for L be Lipschitzian LinearOperator of
  S,T holds R*L is RestFunc of S,U
proof
  let R be RestFunc of T,U such that
A1: R/.0.T=0.U;
  let L be Lipschitzian LinearOperator of S,T;
  consider K be Real such that
A2: 0 <= K and
A3: for h be Point of S holds ||.L.h.|| <= K * ||.h.|| by LOPBAN_1:def 8;
A4: dom L = the carrier of S by FUNCT_2:def 1;
  R is total by NDIFF_1:def 5;
  then dom R = the carrier of T by PARTFUN1:def 2;
  then
A5: rng L c= dom R;
A6: 0 + K < 1 + K by XREAL_1:8;
A7: now
    let e be Real;
    assume e > 0;
    then
A8: 0/(1 + K) < e/(1 + K) by A2,XREAL_1:74;
    set e1=e/(1 + K);
    consider d be Real such that
A9: 0 < d and
A10: for h be Point of T st ||.h.|| < d holds ||.R/.h.|| <= e1*||.h.||
    by A1,A8,Th7;
    set d1=d/( 1 + K );
A11: now
      let h be Point of S such that
A12:  h <> 0.S and
A13:  ||.h.|| < d1;
A14:  ||.L.h.|| <= K * ||.h.|| by A3;
A15:  ||.h.|| <> 0 by A12,NORMSP_0:def 5;
      then
A16:  ||.h.|| > 0 by NORMSP_1:4;
      then K * ||.h.|| < ( K +1) * ||.h.|| by A6,XREAL_1:68;
      then
A17:  ||.L.h.|| < ( K +1) * ||.h.|| by A14,XXREAL_0:2;
      ( K +1) * ||.h.|| < ( K +1) *d1 by A2,A13,XREAL_1:68;
      then ||.L.h.|| < ( K +1) * d1 by A17,XXREAL_0:2;
      then ||.L.h.|| < d by A2,XCMPLX_1:87;
      then
A18:  ||.R/.(L.h).|| <= e1*||.L.h.|| by A10;
A19:  R/.(L.h) = R/.(L/.h) .=(R*L)/.h by A4,A5,PARTFUN2:5;
      e1*||.L.h.|| < e1* ((K +1) * ||.h.||) by A8,A17,XREAL_1:68;
      then ||.R/.(L.h).|| < e1* ((K +1) * ||.h.||) by A18,XXREAL_0:2;
      then
      ||.h.||"* ||.(R*L)/.h.|| < ||.h.||"* (e1* ( K +1) * ||.h.||) by A19,A16,
XREAL_1:68;
      then ||.h.||"* ||.(R*L)/.h.|| < ||.h.||*||.h.||"*e1* ( K +1);
      then ||.h.||"* ||.(R*L)/.h.|| < 1*e1* ( K +1) by A15,XCMPLX_0:def 7;
      hence ||.h.||"* ||.(R*L)/.h.|| < e by A2,XCMPLX_1:87;
    end;
    0/(1 + K) < d/(1 + K) by A2,A9,XREAL_1:74;
    hence
    ex d1 be Real st d1 > 0 &
    for h be Point of S st h <> 0.S & ||.h.|| <
    d1 holds ||.h.||"* ||.(R*L)/.h.|| < e by A11;
  end;
  dom(R*L) = dom L by A5,RELAT_1:27
    .= the carrier of S by FUNCT_2:def 1;
  then R*L is total by PARTFUN1:def 2;
  hence thesis by A7,NDIFF_1:23;
end;
