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

theorem Th29:
  for M be Reflexive symmetric non empty MetrStruct for r be Real
  st r > 0 ex A be Subset of M st (for p,q be Point of M st p <> q & p in A & q
in A holds dist(p,q) >= r) & for p be Point of M ex q be Point of M st q in A &
  p in Ball(q,r)
proof
  let M be Reflexive symmetric non empty MetrStruct;
  let r be Real such that
A1: r > 0;
  set cM=the carrier of M;
  defpred P[set,set] means for p,q be Point of M st p=$1 & q=$2 holds dist(p,q
  )>=r;
A2: for x be Element of cM holds not P[x,x]
  proof
    let x be Element of cM;
    dist(x,x)=0 by METRIC_1:1;
    hence thesis by A1;
  end;
A3: for x,y be Element of cM holds P[x,y] iff P[y,x];
  consider A be Subset of cM such that
A4: for x,y be Element of cM st x in A & y in A & x <> y holds P[x,y] and
A5: for x be Element of cM ex y be Element of cM st y in A & not P[x,y]
  from Th39(A3,A2);
  take A;
  thus for p,q be Point of M st p <> q & p in A & q in A holds dist(p,q) >= r
  by A4;
  let p be Point of M;
  consider y be Element of cM such that
A6: y in A and
A7: not P[p,y] by A5;
  take y;
  thus thesis by A6,A7,METRIC_1:11;
end;
