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 Th11:
  for R1 be RestFunc of S,T st R1/.0.S=0.T for R2 be RestFunc of T,U st R2
/.0.T=0.U for L be Lipschitzian LinearOperator of S,T holds
R2*(L+R1) is RestFunc of S,U
proof
  let R1 be RestFunc of S,T;
  assume R1/.0.S=0.T;
  then consider d0 be Real such that
A1: 0 < d0 and
A2: for h be Point of S st ||.h.|| < d0 holds ||.R1/.h.|| <=1* ||.h.|| by Th7;
  let R2 be RestFunc of T,U such that
A3: R2/.0.T=0.U;
  let L be Lipschitzian LinearOperator of S,T;
  consider K be Real such that
A4: 0 <= K and
A5: for h be Point of S holds ||.L.h.|| <= K * ||.h.|| by LOPBAN_1:def 8;
  R2 is total by NDIFF_1:def 5;
  then dom R2 = the carrier of T by PARTFUN1:def 2;
  then
A6: rng (L+R1) c= dom R2;
  R1 is total by NDIFF_1:def 5;
  then
A7: L+R1 is total by VFUNCT_1:32;
  then
A8: dom(L+R1)=the carrier of S by PARTFUN1:def 2;
A9: now
    let ee be Real such that
A10: ee > 0;
    set e=ee/2;
A11: e < ee by A10,XREAL_1:216;
    set e1=e/( 1 + K );
    e > 0 by A10,XREAL_1:215;
    then 0/(1 + K) < e/(1 + K) by A4,XREAL_1:74;
    then consider d be Real such that
A12: 0 < d and
A13: for z be Point of T st ||.z.|| < d holds ||.R2/.z.|| <= e1*||.z
    .|| by A3,Th7;
    set d1=d/( 1 + K );
    set dd1=min(d0,d1);
A14: dd1 <=d1 by XXREAL_0:17;
A15: dd1 <=d0 by XXREAL_0:17;
A16: now
      let h be Point of S such that
A17:  h <> 0.S and
A18:  ||.h.|| < dd1;
      ||.h.|| < d0 by A15,A18,XXREAL_0:2;
      then
A19:  ||.R1/.h.|| <=1* ||.h.|| by A2;
      ||.L.h.|| <= K * ||.h.|| by A5;
      then ||.L.h+R1/.h.|| <= ||.L.h.|| + ||.R1/.h.|| & ||.L.h.|| + ||.R1/.h
      .|| <= K * ||.h.|| + 1* ||.h.|| by A19,NORMSP_1:def 1,XREAL_1:7;
      then
A20:  ||.L.h+R1/.h.|| <= ( K +1) * ||.h.|| by XXREAL_0:2;
      ||.h.|| < d1 by A14,A18,XXREAL_0:2;
      then ( K +1) * ||.h.|| < ( K +1) *d1 by A4,XREAL_1:68;
      then ||.L.h+R1/.h.|| < ( K +1) * d1 by A20,XXREAL_0:2;
      then ||.L.h+R1/.h.|| < d by A4,XCMPLX_1:87;
      then
A21:  ||.R2/.(L.h+R1/.h).|| <= e1*||.L.h+R1/.h.|| by A13;
      e1*||.L.h+R1/.h.|| <= e1* ((K +1) * ||.h.||) by A4,A10,A20,XREAL_1:64;
      then
A22:  ||.R2/.(L.h+R1/.h).|| <= e1* ((K +1) * ||.h.||) by A21,XXREAL_0:2;
A23:  R2/.(L.h+R1/.h) = R2/.(L/.h+R1/.h) .=R2/.((L+R1)/.h) by A8,VFUNCT_1:def 1
        .=(R2*(L+R1))/.h by A8,A6,PARTFUN2:5;
A24:  ||.h.|| <> 0 by A17,NORMSP_0:def 5;
      then ||.h.|| > 0 by NORMSP_1:4;
      then
      ||.h.||"* ||.(R2*(L+R1))/.h.|| <= ||.h.||"* (e1* ( K +1) * ||.h .||
      ) by A23,A22,XREAL_1:64;
      then ||.h.||"* ||.(R2*(L+R1))/.h.|| <= ||.h.||*||.h.||"*e1* ( K +1);
      then ||.h.||"* ||.(R2*(L+R1))/.h.|| <= 1*e1* ( K +1) by A24,
XCMPLX_0:def 7;
      then ||.h.||"* ||.(R2*(L+R1))/.h.|| <= e by A4,XCMPLX_1:87;
      hence ||.h.||"* ||.(R2*(L+R1))/.h.|| < ee by A11,XXREAL_0:2;
    end;
    0/(1 + K) < d/(1 + K) by A4,A12,XREAL_1:74;
    then 0 < dd1 by A1,XXREAL_0:15;
    hence
    ex dd1 be Real st
    dd1 > 0 & for h be Point of S st h <> 0.S & ||.h.||
    < dd1 holds ||.h.||"* ||.(R2*(L+R1))/.h.|| < ee by A16;
  end;
  dom (R2*(L+R1)) = dom(L+R1) by A6,RELAT_1:27
    .=the carrier of S by A7,PARTFUN1:def 2;
  then R2*(L+R1) is total by PARTFUN1:def 2;
  hence thesis by A9,NDIFF_1:23;
end;
