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 Th82:
  seq is bounded & seq1 is subsequence of seq implies seq1 is bounded
proof
  assume that
A1: seq is bounded and
A2: seq1 is subsequence of seq;
  consider Nseq such that
A3: seq1 = seq * Nseq by A2,VALUED_0:def 17;
  consider M1 such that
A4: M1 > 0 and
A5: for n holds ||.seq.n.|| <= M1 by A1;
  take M = M1;
  now
    let n;
 n in NAT by ORDINAL1:def 12;
    then seq1.n = seq.(Nseq.n) by A3,FUNCT_2:15;
    hence ||.seq1.n.|| <= M by A5;
  end;
  hence thesis by A4;
end;
