reserve F for RealNormSpace;
reserve G for RealNormSpace;
reserve X for set;
reserve x,x0,g,r,s,p for Real;
reserve n,m,k for Element of NAT;
reserve Y for Subset of REAL;
reserve Z for open Subset of REAL;
reserve s1,s3 for Real_Sequence;
reserve seq for sequence of G;
reserve f,f1,f2 for PartFunc of REAL,the carrier of F;
reserve h for 0-convergent non-zero Real_Sequence;
reserve c for constant Real_Sequence;
reserve R,R1,R2 for RestFunc of F;
reserve L,L1,L2 for LinearFunc of F;

theorem Th8:
  r(#)R is RestFunc of F
  proof
    A1: R is total by Def1; then
    A2:r(#)R is total by VFUNCT_1:34;
    now let h;
      dom R = REAL by A1,FUNCT_2:def 1; then
      rng h c= dom R; then
      A3: (h")(#)((r(#)R)/*h) = (h")(#)(r*(R/*h)) by NFCONT_3:4
      .= r*((h")(#)(R/*h)) by NDIFF_1:10;
      A4: (h")(#)(R/*h) is convergent by Def1;
      hence (h")(#)((r(#)R)/*h) is convergent by A3,NORMSP_1:22;
      lim ((h")(#)(R/*h)) = 0.F by Def1;
      hence lim ((h")(#)((r(#)R)/*h)) = r*0.F by A4,A3,NORMSP_1:28
      .= 0.F by RLVECT_1:10;
    end;
    hence thesis by A2,Def1;
  end;
