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 Th9:
  for R be RestFunc of S,T for L be Lipschitzian LinearOperator of T,U holds
  L*R is RestFunc of S,U
proof
  let R be RestFunc of S,T;
  let L be Lipschitzian LinearOperator of T,U;
  consider K be Real such that
A1: 0 <= K and
A2: for z be Point of T holds ||.L.z.|| <= K * ||.z.|| by LOPBAN_1:def 8;
  dom L = the carrier of T by FUNCT_2:def 1;
  then
A3: rng R c= dom L;
A4: R is total by NDIFF_1:def 5;
  then
A5: dom R = the carrier of S by PARTFUN1:def 2;
A6: 0 + K < 1 + K by XREAL_1:8;
A7: now
    let ee be Real such that
A8: ee > 0;
    set e=ee/2;
    e > 0 by A8,XREAL_1:215;
    then
A9: 0/(1 + K) < e/(1 + K) by A1,XREAL_1:74;
    set e1=e/( 1 + K );
    R is total by NDIFF_1:def 5;
    then consider d be Real such that
A10: 0 < d and
A11: for h be Point of S st h <> 0.S & ||.h.|| < d holds ||.h.||"* ||.
    R/.h.|| < e1 by A9,NDIFF_1:23;
A12: e < ee by A8,XREAL_1:216;
    now
      let h be Point of S such that
A13:  h <> 0.S and
A14:  ||.h.|| < d;
      ||.h.||"* ||.(R/.h).|| < e1 by A11,A13,A14;
      then ( K +1) *( ||.h.||"* ||.R/.h.||) <=( K +1) *e1 by A1,XREAL_1:64;
      then
A15:  ( K +1) *( ||.h.||"* ||.R/.h.||) <=e by A1,XCMPLX_1:87;
      ||.h.|| <> 0 by A13,NORMSP_0:def 5;
      then
A16:  ||.h.|| > 0 by NORMSP_1:4;
      0 <= ||.R/.h.|| by NORMSP_1:4;
      then
A17:  K * ||.R/.h.|| <= ( K +1) * ||.R/.h.|| by A6,XREAL_1:64;
      ||.L.(R/.h).|| <= K * ||.R/.h.|| by A2;
      then ||.L.(R/.h).|| <= ( K +1) * ||.R/.h.|| by A17,XXREAL_0:2;
      then
      ||.h.||"* ||.L.(R/.h).|| <= ||.h.||"*(( K +1) * ||.R/.h.|| ) by A16,
XREAL_1:64;
      then
A18:  ||.h.||"* ||.L.(R/.h).|| <= e by A15,XXREAL_0:2;
      L.(R/.h) = L/.(R/.h) .=(L*R)/.h by A5,A3,PARTFUN2:5;
      hence ||.h.||"* ||.(L*R)/.h.|| < ee by A12,A18,XXREAL_0:2;
    end;
    hence
    ex d be Real st d > 0 &
    for h be Point of S st h <> 0.S & ||.h.|| < d
    holds ||.h.||"* ||.(L*R)/.h.|| < ee by A10;
  end;
  dom(L*R) = dom R by A3,RELAT_1:27
    .= the carrier of S by A4,PARTFUN1:def 2;
  then L*R is total by PARTFUN1:def 2;
  hence thesis by A7,NDIFF_1:23;
end;
