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

theorem Th2:
  for M be Reflexive non empty MetrStruct for S be pointwise_bounded
  SetSequence of M st S is non-ascending holds diameter S is bounded_above &
  diameter S is non-increasing
proof
  let M be Reflexive non empty MetrStruct;
  let S be pointwise_bounded SetSequence of M such that
A1: S is non-ascending;
  set d=diameter S;
A2: now
    let n be Nat;
A3: d.0+0<d.0+1 by XREAL_1:8;
A4: diameter S.n=d.n by Def2;
A5: diameter S.0=d.0 by Def2;
A6: S.0 is bounded by Def1;
    S.n c= S.0 by A1,PROB_1:def 4;
    then d.n<=d.0 by A6,A4,A5,TBSP_1:24;
    hence d.n<d.0+1 by A3,XXREAL_0:2;
  end;
  now
    let m,n be Nat such that
    m in dom d and
    n in dom d and
A7: m <= n;
A8: S.m is bounded by Def1;
A9: diameter S.m=d.m by Def2;
A10: diameter S.n=d.n by Def2;
    S.n c= S.m by A1,A7,PROB_1:def 4;
    hence d.n<=d.m by A8,A10,A9,TBSP_1:24;
  end;
  hence thesis by A2,SEQM_3:def 4,SEQ_2:def 3;
end;
