reserve F for RealNormSpace;
reserve G for RealNormSpace;
reserve y,X for set;
reserve x,x0,x1,x2,g,g1,g2,r,r1,s,p,p1,p2 for Real;
reserve i,m,k for Element of NAT;
reserve n,k for non zero Element of NAT;
reserve Y for Subset of REAL;
reserve Z for open Subset of REAL;
reserve s1,s3 for Real_Sequence;
reserve seq,seq1 for sequence of G;
reserve f,f1,f2 for PartFunc of REAL,REAL n;
reserve g,g1,g2 for PartFunc of REAL,REAL-NS n;
reserve h for 0-convergent non-zero Real_Sequence;
reserve c for constant Real_Sequence;
reserve GR,R for RestFunc of REAL-NS n;
reserve DFG,L for LinearFunc of REAL-NS n;
reserve m for non zero Element of NAT;

theorem Th47:
  for R be RestFunc of REAL-NS n
  for L be Lipschitzian LinearOperator of REAL-NS n,REAL-NS m holds
  L*R is RestFunc of REAL-NS m
proof
  let R be RestFunc of REAL-NS n;
  let L be Lipschitzian LinearOperator of REAL-NS n,REAL-NS m;
  set S=REAL-NS n;
  set T=REAL-NS m;
  consider K be Real such that
A1: 0 <= K and
A2: for z be Point of S holds ||.L.z.|| <= K * ||.z.|| by LOPBAN_1:def 8;
  dom L = the carrier of S by FUNCT_2:def 1;
  then
A3: rng R c= dom L;
A4: R is total by NDIFF_3:def 1;
  then
A5: dom R = REAL by PARTFUN1:def 2;
  reconsider p0=0, p1=1 as Real;
A6: p0 + K < p1 + K by XREAL_1:8;
now
    let ee be Real such that
A7: ee > 0;
    set e=ee/2;
    e > 0 by A7,XREAL_1:215;
    then
A8: 0/(1 + K) < e/(1 + K) by A1,XREAL_1:74;
    set e1=e/( 1 + K );
    consider d be Real such that
A9: 0 < d and
A10: for h be Real st h <> 0 & |.h.| < d holds (|.h.|"* ||.
    R/.h.||) < e1 by A4,A8,Th23;
A11: e < ee by A7,XREAL_1:216;
    now
      let h be Real such that
A12:  h <> 0 and
A13:  |.h.| < d;
       reconsider hh=h as Element of REAL by XREAL_0:def 1;
      |.h.|"* ||.(R/.h).|| < e1 by A10,A12,A13;
      then ( K +1) *( |.h.|"* ||.R/.h.||) <=( K +1) *e1 by A1,XREAL_1:64;
      then
A14:  ( K +1) *( |.h.|"* ||.R/.h.||) <=e by A1,XCMPLX_1:87;
      |.h.| <> 0 by A12,COMPLEX1:45;
      then
A15:  |.h.| > 0 by COMPLEX1:46;
A16:  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 A16,XXREAL_0:2;
      then
      |.h.|"* ||.L/.(R/.h).|| <= |.h.|"*(( K +1) * ||.R/.h.|| ) by A15,
XREAL_1:64;
      then
A17:  |.h.|"* ||.L/.(R/.h).|| <= e by A14,XXREAL_0:2;
      L/.(R/.h) = L/.(R/.hh) .=(L*R)/.hh by A5,A3,PARTFUN2:5;
      hence |.h.|"* ||.(L*R)/.h.|| < ee by A11,A17,XXREAL_0:2;
    end;
    hence ex d be Real st d > 0 &
     for h be Real st h <> 0 & |.h.| < d
    holds |.h.|"* ||.(L*R)/.h.|| < ee by A9;
  end;
  hence thesis by A4,Th23;
end;
