 reserve j for set;
 reserve p,r for Real;
 reserve S,T,F for RealNormSpace;
 reserve x0 for Point of S;
 reserve g for PartFunc of S,T;
 reserve c for constant sequence of S;
 reserve R for RestFunc of S,T;
 reserve G for RealNormSpace-Sequence;
 reserve i for Element of dom G;
 reserve f for PartFunc of product G,F;
 reserve x for Element of product G;

theorem Th3:
  for R be RestFunc of S
  for L be Lipschitzian LinearOperator of S,T holds
    L*R is RestFunc of T
proof
   let R be RestFunc of S;
   let L be Lipschitzian LinearOperator of S,T;
   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;
   now let e be Real such that
A6:  e > 0;
     set e1 = e/2/(1 + K);
     consider d be Real such that
A7:   0 < d and
A8:   for h be Real st h <> 0 & |.h.| < d
        holds |.h.|"* ||. R/.h .|| < e1
         by A1,A4,A6,Th1;
A9: e/2 < e by A6,XREAL_1:216;
     now let h be Real;
       reconsider hh=h as Element of REAL by XREAL_0:def 1;
       assume A10: h <> 0 & |.h.| < d; then
       |.h.|"* ||.(R/.h).|| < e1 by A8; then
       (K +1)*(|.h.|"* ||.R/.h.||) <= (K +1)*e1 by A1,XREAL_1:64; then
A11:   (K +1)*(|.h.|"* ||.R/.h.||) <= e/2 by A1,XCMPLX_1:87;
       |.h.| <> 0 by A10,COMPLEX1:45; then
A12:   |.h.| > 0 by COMPLEX1:46;
       reconsider p0=0, p1=1 as Element of REAL by XREAL_0:def 1;
       p0 + K < p1 + K by XREAL_1:8; then
A13:   K * ||.R/.h.|| <= (K +1) * ||.R/.h.|| by XREAL_1:64;
       ||.L.(R/.h).|| <= K * ||.R/.h.|| by A2; then
       ||.L.(R/.h).|| <= (K +1) * ||.R/.h.|| by A13,XXREAL_0:2; then
       |.h.|"* ||.L.(R/.h).|| <= |.h.|"*((K +1)*||. R/.h .||) by A12,XREAL_1:64
; then
A14:   |.h.|"* ||.L.(R/.h).|| <= e/2 by A11,XXREAL_0:2;
       L.(R/.h) = L/.(R/.h); then
       L.(R/.hh) =(L*R)/.hh by A5,A3,PARTFUN2:5;
       hence |.h.|"* ||.(L*R)/.h.|| < e by A9,A14,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.|| < e by A7;
   end;
   hence thesis by A4,Th1;
end;
