reserve n,k for Element of NAT;
reserve x,y,X for set;
reserve g,r,p for Real;
reserve S for RealNormSpace;
reserve rseq for Real_Sequence;
reserve seq,seq1 for sequence of S;
reserve x0 for Point of S;
reserve Y for Subset of S;
reserve S,T for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve s1 for sequence of S;
reserve x0 for Point of S;
reserve h for (0.S)-convergent sequence of S;
reserve c for constant sequence of S;

theorem Th24:
  for R be RestFunc of S,T for s be (0.S)-convergent sequence of S
   st s is non-zero
  holds (R/*s) is convergent & lim (R/*s) = 0.T
proof
  let R be RestFunc of S,T;
  let s be (0.S)-convergent sequence of S
  such that A1: s is non-zero ;
A2: (||.s.||") (#) (R/*s) is convergent by Def5,A1;
  now
    let n be Element of NAT;
    s.n <> 0.S by Th7,A1;
    then
A3: ||.s.n.|| <> 0 by NORMSP_0:def 5;
    thus (||.s.||(#) ( (||.s.||") (#) (R/*s) )).n =||.s.||.n*((||.s.||")(#)(R
    /*s)).n by Def2
      .=||.s.||.n*((||.s.||").n*((R/*s).n)) by Def2
      .= ||.s.n.||*((||.s.||").n*((R/*s).n)) by NORMSP_0:def 4
      .= ||.s.n.||*((||.s.||.n)"*((R/*s).n)) by VALUED_1:10
      .= ||.s.n.||*((||.s.n.||)"*((R/*s).n)) by NORMSP_0:def 4
      .= ||.s.n.||*(||.s.n.||)"*((R/*s).n) by RLVECT_1:def 7
      .= 1*(R/*s).n by A3,XCMPLX_0:def 7
      .=(R/*s).n by RLVECT_1:def 8;
  end;
  then
A4: ||.s.||(#) ( (||.s.||") (#) (R/*s) )= (R/*s) by FUNCT_2:63;
A5: s is convergent by Def4;
  then
A6: ||.s.|| is convergent by LOPBAN_1:41;
  lim s = 0.S by Def4;
  then lim ( ||.s.|| ) = ||.0.S.|| by A5,LOPBAN_1:41;
  then
A7: lim ( ||.s.|| ) =0 by NORMSP_1:1;
  lim ((||.s.||") (#) (R/*s) ) = 0.T by Def5,A1;
  then lim (R/*s) = 0* 0.T by A5,A4,A2,A7,Th14,LOPBAN_1:41;
  hence thesis by A4,A2,A6,Th13,RLVECT_1:10;
end;
