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 Th74:
  seq1 is bounded & seq2 is bounded implies seq1 + seq2 is bounded
proof
  assume that
A1: seq1 is bounded and
A2: seq2 is bounded;
  consider M2 such that
A3: M2 > 0 and
A4: for n holds ||.seq2.n.|| <= M2 by A2;
  consider M1 such that
A5: M1 > 0 and
A6: for n holds ||.seq1.n.|| <= M1 by A1;
  now
    let n;
    ||.(seq1 + seq2).n.|| = ||.seq1.n + seq2.n.|| by NORMSP_1:def 2;
    then
A7: ||.(seq1 + seq2).n.|| <= ||.seq1.n.|| + ||.seq2.n.|| by CSSPACE:46;
    ||.seq1.n.|| <= M1 & ||.seq2.n.|| <= M2 by A6,A4;
    then ||.seq1.n.|| + ||.seq2.n.|| <= M1 + M2 by XREAL_1:7;
    hence ||.(seq1 + seq2).n.|| <= M1 + M2 by A7,XXREAL_0:2;
  end;
  hence thesis by A5,A3;
end;
