reserve i,n,m for Nat,
  x,y,X,Y for set,
  r,s for Real;

theorem Th18:
  for M be non empty MetrSpace, A be non empty Subset of M for S
be sequence of (M|A),S9 be sequence of M st S = S9 holds S9 is Cauchy iff S is
  Cauchy
proof
  let M be non empty MetrSpace, A be non empty Subset of M;
  let S be sequence of (M|A),S9 be sequence of M such that
A1: S = S9;
  thus S9 is Cauchy implies S is Cauchy
  proof
    assume
A2: S9 is Cauchy;
    let r;
    assume r>0;
    then consider p be Nat such that
A3: for n,m be Nat st p<=n & p<=m holds dist(S9.n,S9.m)<r
    by A2;
    take p;
    let n,m be Nat such that
A4: p<=n and
A5: p<=m;
    dist(S.n,S.m) = dist(S9.n,S9.m) by A1,TOPMETR:def 1;
    hence thesis by A3,A4,A5;
  end;
  assume
A6: S is Cauchy;
  let r;
  assume r>0;
  then consider p be Nat such that
A7: for n,m be Nat st p<=n & p<=m holds dist(S.n,S.m)<r by A6;
  take p;
  let n,m be Nat such that
A8: p <= n and
A9: p <= m;
  dist(S.n,S.m) = dist(S9.n,S9.m) by A1,TOPMETR:def 1;
  hence thesis by A7,A8,A9;
end;
