reserve X for RealUnitarySpace;
reserve x, y, z, g, g1, g2 for Point of X;
reserve a, q, r for Real;
reserve seq, seq1, seq2, seq9 for sequence of X;
reserve k, n, m, m1, m2 for Nat;

theorem Th10:
  seq is constant & x in rng seq implies lim seq = x
proof
  assume that
A1: seq is constant and
A2: x in rng seq;
  consider y such that
A3: rng seq = {y} by A1,FUNCT_2:111;
  consider z such that
A4: for n being Nat holds seq.n = z by A1,VALUED_0:def 18;
A5: x = y by A2,A3,TARSKI:def 1;
A6: now
    let r such that
A7: r > 0;
     reconsider m = 0 as Nat;
    take m;
    let n such that
    n >= m;
  n in NAT by ORDINAL1:def 12;
    then n in dom seq by NORMSP_1:12;
    then seq.n in rng seq by FUNCT_1:def 3;
    then z in rng seq by A4;
    then z = x by A3,A5,TARSKI:def 1;
    then seq.n = x by A4;
    hence dist((seq.n) , x) < r by A7,BHSP_1:34;
  end;
  seq is convergent by A1;
  hence thesis by A6,Def2;
end;
