reserve X for MetrSpace,
  x,y,z for Element of X,
  A for non empty set,
  G for Function of [:A,A:],REAL,
  f for Function,
  k,n,m,m1,m2 for Nat,
  q,r for Real;
reserve X for non empty MetrSpace,
  x,y for Element of X,
  V for Subset of X,
  S,S1,T for sequence of X,
  Nseq for increasing sequence of NAT;

theorem
  S is_convergent_in_metrspace_to x & S is_convergent_in_metrspace_to y
  implies x = y
proof
  assume that
A1: S is_convergent_in_metrspace_to x and
A2: S is_convergent_in_metrspace_to y;
A3: lim S = y by A2,Th12;
A4: for n being Nat holds sequence_of_dist(S,S).n = In(0,REAL)
  proof
    let n be Nat;
    sequence_of_dist(S,S).n = dist(S.n,S.n) by Def7
      .= 0 by METRIC_1:1;
    hence thesis;
  end;
A5: ex n st sequence_of_dist(S,S).n = 0
  proof
    take 0;
    thus thesis by A4;
  end;
  lim S = x & S is convergent by A1,Th12;
  then
A6: dist(x,y) = lim sequence_of_dist(S,S) by A3,Th21;
  sequence_of_dist(S,S) is constant by A4,VALUED_0:def 18;
  then dist(x,y) = 0 by A6,A5,SEQ_4:25;
  hence thesis by METRIC_1:2;
end;
