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

theorem Th4:
  for M be non empty Reflexive MetrStruct for S be pointwise_bounded
  SetSequence of M st S is non-ascending & lim diameter S = 0 for F be sequence
  of M st for i holds F.i in S.i holds F is Cauchy
proof
  let M be non empty Reflexive MetrStruct;
  let S be pointwise_bounded SetSequence of M such that
A1: S is non-ascending and
A2: lim diameter S = 0;
  set d=diameter S;
A3: d is non-increasing by A1,Th2;
A4: d is bounded_below by Th1;
  let F be sequence of M such that
A5: for i holds F.i in S.i;
  let r;
  assume r>0;
  then consider n be Nat such that
A6: for m be Nat st n <= m holds |.d.m-0 .| < r by A2,A4,A3,
SEQ_2:def 7;
  take n;
  let m1,m2 be Nat such that
A7: n <= m1 and
A8: n <= m2;
A9: S.m2 c= S.n by A1,A8,PROB_1:def 4;
A10: F.m2 in S.m2 by A5;
A11: F.m1 in S.m1 by A5;
A12: |.d.n-0 .|<r by A6;
A13: diameter S.n=d.n by Def2;
A14: S.n is bounded by Def1;
  then 0 <= d.n by A13,TBSP_1:21;
  then
A15: d.n < r by A12,ABSVALUE:def 1;
  S.m1 c= S.n by A1,A7,PROB_1:def 4;
  then dist(F.m1,F.m2)<=d.n by A9,A11,A10,A14,A13,TBSP_1:def 8;
  hence thesis by A15,XXREAL_0:2;
end;
