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;

theorem Th46:
  for R be RestFunc of REAL-NS n st R/.0=0.(REAL-NS n)
  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 REAL-NS n such that
A1: R/.0=0.(REAL-NS n);
  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,Th23;
  take d;
  now
    let h be Real such that
A5: |.h.| < d;
    now
      per cases;
      case
A6:     h <> 0;
        then 0 <= |.h.| & |.h.|"*||. R/.h .|| <= e by A4,A5,COMPLEX1:46;
        then |.h.|*(|.h.|"*||. R/.h .||) <= |.h.|*e by XREAL_1:64;
        then
A7:     |.h.|*|.h.|"*||. R/.h .|| <= e* |.h.|;
        |.h.| <> 0 by A6,COMPLEX1:45;
        then 1*||. R/.h .|| <= e* |.h.| by A7,XCMPLX_0:def 7;
        hence ||. R/.h .|| <= e* |.h.|;
      end;
      case
A8:     h = 0;
        reconsider p0=0 as Real;
        0 <= |.h.| by COMPLEX1:46;
        then p0* |.h.| <= e* |.h.| by A2;
        hence ||. R/.h .|| <= e* |.h.| by A1,A8;
      end;
    end;
    hence ||. R/.h .|| <= e* |.h.|;
  end;
  hence thesis by A3;
end;
