
theorem Th7:
  the_set_of_l2ComplexSequences is linearly-closed
proof
  set W = the_set_of_l2ComplexSequences;
  hereby
    let v,u be VECTOR of Linear_Space_of_ComplexSequences such that
A1: v in W and
A2: u in W;
    |.seq_id(v+u).|(#)|.seq_id(v+u).| is summable
    proof
      set r = |.seq_id(v+u).|(#)|.seq_id(v+u).|;
      set q = |.seq_id(u).|(#)|.seq_id(u).|;
      set p = |.seq_id(v).|(#)|.seq_id(v).|;
A3:   for n be Nat holds 0<=r.n
      proof
        let n be Nat;
        r.n=(|.seq_id(v+u).|).n * (|.seq_id(v+u).|).n by SEQ_1:8;
        hence thesis by XREAL_1:63;
      end;
A4:   for n be Nat holds r.n <=(2(#)p+2(#)q).n
      proof
        set t = |.seq_id(u).|;
        set s = |.seq_id(v).|;
        let n be Nat;
A5:      n in NAT by ORDINAL1:def 12;
        reconsider sn=s.n, tn=t.n as Real;
A6:     r.n = ((|.seq_id(v+u).|).n)^2 by SEQ_1:8;
        (2(#)p+2(#)q).n=(2(#)p).n +(2(#)q).n by SEQ_1:7
          .= 2*p.n + (2(#)q).n by SEQ_1:9
          .= 2*p.n + 2*q.n by SEQ_1:9
          .= 2*(s.n*s.n) + 2*q.n by SEQ_1:8
          .= 2*sn^2 + 2*tn^2 by SEQ_1:8;
        then
A7:     (2(#)p+2(#)q).n - (sn^2 + 2*sn*tn + tn^2) = (sn-tn)^2;
A8:     v+u=seq_id(v)+seq_id(u) by Th2;
        (|.seq_id(v+u).|).n = |.(seq_id(v+u)).n .| by VALUED_1:18
          .= |.(seq_id(v)).n + (seq_id(u)).n.| by A8,VALUED_1:1,A5;
        then (|.seq_id(v+u).|).n <= |.(seq_id(v)).n.| + |.(seq_id(u)).n.| by
COMPLEX1:56;
        then (|.seq_id(v+u).|).n <= s.n + |.((seq_id(u)).n).| by VALUED_1:18;
        then
A9:     (|.seq_id(v+u).|).n <= s.n + t.n by VALUED_1:18;
A10:    0 + (sn^2 + 2*sn*tn + tn^2) <= (2(#)p+2(#)q).n by A7,XREAL_1:19,63;
        0 <= |.(seq_id(v+u)).n.| by COMPLEX1:46;
        then 0 <= (|.seq_id(v+u).|).n by VALUED_1:18;
        then ((|.seq_id(v+u).|).n)^2 <= (s.n + t.n)^2 by A9,SQUARE_1:15;
        hence thesis by A6,A10,XXREAL_0:2;
      end;
      |.seq_id(u).|(#)|.seq_id(u).| is summable by A2,Def9;
      then
A11:  2(#)q is summable by SERIES_1:10;
      |.seq_id(v).|(#)|.seq_id(v).| is summable by A1,Def9;
      then 2(#)p is summable by SERIES_1:10;
      then 2(#)p+2(#)q is summable by A11,SERIES_1:7;
      hence thesis by A3,A4,SERIES_1:20;
    end;
    hence v+u in W by Def9;
  end;
    let z be Complex;
    let v be VECTOR of Linear_Space_of_ComplexSequences;
    assume v in W;
    then
A12: |.seq_id(v).|(#)|.seq_id(v).| is summable by Def9;
    z*v=z(#)seq_id(v) by Th3;
    then |.seq_id(z*v).| = |.z.|(#)|.seq_id(v).| by COMSEQ_1:50;
    then |.seq_id(z*v).|(#)|.seq_id(z*v).| =|.z.|(#)(|.z.|(#)|.seq_id(v).|(#)
    |.seq_id(v).|) by SEQ_1:18
      .=|.z.|(#)(|.z.|(#)(|.seq_id(v).|(#)|.seq_id(v).|)) by SEQ_1:18
      .=(|.z.|*|.z.|)(#)(|.seq_id(v).|(#)|.seq_id(v).|) by SEQ_1:23;
    then |.seq_id(z*v).|(#)|.seq_id(z*v).| is summable by A12,SERIES_1:10;
    hence thesis by Def9;
end;
