reserve n for Nat;

theorem Th18:
  for n be Nat, seq1 be sequence of REAL-NS n, seq2 be
  sequence of REAL-US n st seq1 = seq2 & seq2 is Cauchy holds seq1 is
  Cauchy_sequence_by_Norm
proof
  let n be Nat, seq1 be sequence of REAL-NS n, seq2 be sequence of
  REAL-US n;
  assume that
A1: seq1 = seq2 and
A2: seq2 is Cauchy;
  let r be Real;
  assume r > 0;
  then consider k be Nat such that
A3: for m1,m2 be Nat st m1 >= k & m2 >= k holds dist(seq2.m1,
  seq2.m2) < r by A2;
  take k;
  let m1,m2 be Nat;
  reconsider p = seq2.m1 - seq2.m2 as Element of REAL n by Def6;
  -seq1.m2 = -seq2.m2 by A1,Th13;
  then
A4: p = seq1.m1 - seq1.m2 by A1,Th13;
  assume m1 >= k & m2 >= k;
  then
A5: dist(seq2.m1, seq2.m2) < r by A3;
  ||. seq2.m1 - seq2.m2 .|| = sqrt ((Euclid_scalar n). (p,p)) by Def6
    .= sqrt Sum mlt(p,p) by Def5
    .= sqrt |(p,p)| by RVSUM_1:def 16
    .= |.p.| by EUCLID_2:5
    .= ||. seq1.m1 - seq1.m2 .|| by A4,Th1;
  hence thesis by A5;
end;
