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

theorem
  for M be Reflexive non empty MetrStruct
  for S be pointwise_bounded SetSequence of M st
  S is non-descending holds diameter S is non-decreasing
proof
  let M be Reflexive non empty MetrStruct;
  let S be pointwise_bounded SetSequence of M such that
A1: S is non-descending;
  set d=diameter S;
  now
    let m,n be Nat such that
    m in dom d and
    n in dom d and
A2: m <= n;
A3: S.n is bounded by Def1;
A4: diameter S.m=d.m by Def2;
A5: diameter S.n=d.n by Def2;
    S.m c= S.n by A1,A2,PROB_1:def 5;
    hence d.n>=d.m by A3,A5,A4,TBSP_1:24;
  end;
 hence thesis by SEQM_3:def 3;
end;
