
theorem RHS11a:
  for X be RealUnitarySpace, s1 be sequence of X,
      s2 be sequence of RUSp2RNSp X
  st s1 = s2 holds s2 is Cauchy_sequence_by_Norm iff s1 is Cauchy
proof
  let X be RealUnitarySpace,
      s1 be sequence of X, s2 be sequence of RUSp2RNSp X;
  assume A0: s1 = s2;
  hereby assume AS: s2 is Cauchy_sequence_by_Norm;
   for r be Real st 0 < r ex k be Nat st
     for n, m be Nat st n >= k & m >= k holds ||.s1.n - s1.m.|| < r
   proof
     let r be Real;
     assume 0 < r; then
     consider k be Nat such that
P1:   for n, m be Nat st n >= k & m >= k
        holds ||.s2.n - s2.m.|| < r by AS,RSSPACE3:8;
     take k;
     thus
     for n, m be Nat st n >= k & m >= k holds ||.s1.n - s1.m.|| < r
     proof
      let n, m be Nat;
      assume n >= k & m >= k; then
P2:   ||.s2.n - s2.m.|| < r by P1;
      s2.n - s2.m = s1.n - s1.m by A0,RHS3;
      hence ||.s1.n - s1.m.|| < r by Def1,P2;
     end;
   end;
   hence s1 is Cauchy by BHSP_3:2;
  end;
  assume A1: s1 is Cauchy;
  for r be Real st r > 0 ex k be Nat st for n, m be Nat st n >= k & m >= k
    holds ||.s2.n - s2.m.|| < r
  proof
    let r be Real;
    assume r > 0; then
    consider k be Nat such that
A2:   for n, m be Nat st n >= k & m >= k
        holds ||.s1.n - s1.m.|| < r by A1,BHSP_3:2;
    take k;
    thus for n, m be Nat st n >= k & m >= k holds ||.s2.n - s2.m.|| < r
    proof
      let n, m be Nat;
      assume n >= k & m >= k; then
A3:   ||.s1.n - s1.m.|| < r by A2;
      s1.n - s1.m = s2.n - s2.m by A0,RHS3;
      hence ||.s2.n - s2.m.|| < r by Def1,A3;
    end;
  end;
  hence s2 is Cauchy_sequence_by_Norm by RSSPACE3:8;
end;
