reserve n,m,k,k1,k2 for Nat;
reserve r,r1,r2,s,t,p for Real;
reserve seq,seq1,seq2 for Real_Sequence;
reserve x,y for set;

theorem Th29:
  for seq, n holds
    {seq.k: n <= k} is Subset of REAL
proof
  let seq, n;
  set Y = {seq.k: n <= k};
  now
    let x be object;
    assume x in Y;
    then consider z1 be set such that
A1: z1 = x and
A2: z1 in Y;
    consider k1 being Nat such that
A3: z1 = seq.k1 & n <= k1 by A2;
    reconsider k1 as Element of NAT by ORDINAL1:def 12;
    z1 = seq.k1 by A3;
    hence x in REAL by A1;
  end;
  hence thesis by TARSKI:def 3;
end;
