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

theorem Th1:
  for M be Reflexive non empty MetrStruct for S be pointwise_bounded
  SetSequence of M holds diameter S is bounded_below
proof
  let M be Reflexive non empty MetrStruct;
  let S be pointwise_bounded SetSequence of M;
  set d=diameter S;
  now
    let n be Nat;
A1: diameter (S.n)=d.n by Def2;
    S.n is bounded by Def1;
    then 0<=d.n by A1,TBSP_1:21;
    hence -1 < d.n by XXREAL_0:2;
  end;
  hence thesis by SEQ_2:def 4;
end;
