 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 Th2:
  for R be RestFunc of S st R/.0=0.S
  for e be Real st e > 0 ex d be Real st
    d > 0 & for h be Real st |.h.| < d holds ||.R/.h.|| <= e*|.h.|
proof
   let R be RestFunc of S such that
A1:  R/.0=0.S;
   let e be Real such that
A2:  e > 0;
   R is total by NDIFF_3:def 1; then
   consider d be Real such that
A3:  d > 0 and
A4:  for z be Real st z <> 0 & |.z.| < d
       holds |.z.|"* ||. R/.z .|| < e by A2,Th1;
   take d;
   now let h be Real such that
A5: |.h.| < d;
A6: 0 <= |.h.| by COMPLEX1:46;
    per cases;
    suppose A7: h <> 0; then
     |.h.|"*||. R/.h .|| <= e by A4,A5; then
     |.h.|*(|.h.|"*||. R/.h .||) <= |.h.|*e by A6,XREAL_1:64; then
A8:  |.h.|*|.h.|"*||. R/.h .|| <= e * |.h.|;
     |.h.| <> 0 by A7,COMPLEX1:45; then
     1*||. R/.h .|| <= e * |.h.| by A8,XCMPLX_0:def 7;
     hence ||. R/.h .|| <= e* |.h.|;
    end;
    suppose A9: h = 0;
     reconsider p0=0 as Real;
     p0* |.h.| <= e* |.h.| by A2,A6;
     hence ||. R/.h .|| <= e* |.h.| by A1,A9;
    end;
   end;
   hence thesis by A3;
end;
