
theorem RHS9:
  for X be RealUnitarySpace,
      s1 be sequence of X, s2 be sequence of RUSp2RNSp X
  st s1 = s2 & s1 is convergent holds
   lim s1 = lim s2
proof
  let X be RealUnitarySpace,
      s1 be sequence of X, s2 be sequence of RUSp2RNSp X;
  assume AS: s1 = s2;
  assume P1: s1 is convergent; then
P5: s2 is convergent by AS,RHS8;
  reconsider g1=lim s1 as Point of RUSp2RNSp X;
  now let p be Real;
    assume 0 < p; then
    consider m be Nat such that
P2:   for n be Nat st m <= n holds ||.s1.n - lim s1.|| < p
        by P1,BHSP_2:19;
    take m;
    thus for n be Nat st m <= n holds ||.s2.n - g1.|| < p
    proof
      let n be Nat;
      assume m <= n; then
P3:   ||.s1.n - lim s1.|| < p by P2;
      s1.n - lim s1 = s2.n - g1 by AS,RHS3;
      hence ||.s2.n - g1.|| < p by P3,Def1;
    end;
  end;
  hence lim s2 = lim s1 by P5,NORMSP_1:def 7;
end;
