reserve X for ComplexUnitarySpace;
reserve x, y, w, g, g1, g2 for Point of X;
reserve z for Complex;
reserve p, q, r, M, M1, M2 for Real;
reserve seq, seq1, seq2, seq3 for sequence of X;
reserve k,n,m for Nat;
reserve Nseq for increasing sequence of 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 w such that
A4: for n being Nat holds seq.n = w 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 w in rng seq by A4;
    then w = x by A3,A5,TARSKI:def 1;
    then seq.n = x by A4;
    hence dist((seq.n) , x) < r by A7,CSSPACE:50;
  end;
  seq is convergent by A1,Th1;
  hence thesis by A6,Def2;
end;
