reserve NRM for non empty RealNormSpace;
reserve seq for sequence of NRM;

theorem Th8:
  seq is Cauchy_sequence_by_Norm iff for r be Real st r > 0
    ex k be Nat st for n, m be Nat st n >= k & m >= k holds ||.(
  seq.n) - (seq.m).|| < r
proof
  thus seq is Cauchy_sequence_by_Norm implies
  for r be Real st r > 0
   ex k be Nat st for n, m be Nat st n >= k & m >= k holds ||.(seq.n
  ) - (seq.m).|| < r
  proof
    assume
A1: seq is Cauchy_sequence_by_Norm;
    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 dist(seq.n,
    seq.m) < r by A1;
    for n, m be Nat st n >= k & m >= k holds ||.(seq.n) - (seq.
    m).|| < r
    proof
      let n,m be Nat;
      assume n >= k & m >= k;
      then dist(seq.n, seq.m) < r by A2;
      hence thesis;
    end;
    hence thesis;
  end;
    assume
A3: for r be Real st r > 0 ex k be Nat st for n, m be
    Nat st n >= k & m >= k holds ||.(seq.n) - (seq.m).|| < r;
    now
      let r be Real;
      assume
A4:   r > 0;
      now
        consider k be Nat such that
A5:     for n, m be Nat st n >= k & m >= k holds ||.(seq.n
        ) - (seq.m).|| < r by A3,A4;
        for n,m be Nat st n >= k & m >= k holds dist(seq.n,
        seq.m) < r by A5;
        hence ex k be Nat st for n, m be Nat st n >= k &
        m >= k holds dist(seq.n, seq.m) < r;
      end;
      hence ex k be Nat st for n, m be Nat st n >= k & m
      >= k holds dist(seq.n, seq.m) < r;
    end;
    hence thesis;
end;
