reserve x1,x2,y1,a,b,c for Real;

theorem Th4:
  for p be Real
   st 1 <= p holds the_set_of_RealSequences_l^p is linearly-closed
proof
  let p be Real such that
A1: p >=1;
  set W = the_set_of_RealSequences_l^p;
A2: for v,u be VECTOR of Linear_Space_of_RealSequences st v in
the_set_of_RealSequences_l^p & u in the_set_of_RealSequences_l^p holds v + u in
  the_set_of_RealSequences_l^p
  proof
    let v,u be VECTOR of Linear_Space_of_RealSequences such that
A3: v in W and
A4: u in W;
A5: seq_id(v +u) rto_power p is summable
    proof
      reconsider vq=v as Real_Sequence by FUNCT_2:66;
      set up=seq_id(u) rto_power p;
      set vp=seq_id(v) rto_power p;
      set p1=1/p;
A7:   now
        let n be Nat;
        thus
A8:     vp.n=|.(seq_id(v)).n.| to_power p by Def1;
        thus
A9:     up.n=|.(seq_id(u)).n.| to_power p by Def1;
        thus 0 <= |.(seq_id(v)).n.| by COMPLEX1:46;
        thus 0 < |.(seq_id(v)).n.| or 0=|.(seq_id(v)).n.| by COMPLEX1:46;
        hence 0 <= vp.n by A1,A8,POWER:34,def 2;
        thus 0 <= |.(seq_id(u)).n.| by COMPLEX1:46;
        thus 0 < |.(seq_id(u)).n.| or 0=|.(seq_id(u)).n.| by COMPLEX1:46;
        hence 0 <= up.n by A1,A9,POWER:34,def 2;
      end;
      seq_id(v) rto_power p is summable by A1,A3,Def2;
      then Partial_Sums(vp) is bounded_above by A7,SERIES_1:17;
      then consider r be Real such that
A10:  for n be object st n in dom Partial_Sums(vp) holds Partial_Sums(vp
      ).n<r by SEQ_2:def 1;
A11:  1/p > 0 by A1,XREAL_1:139;
      reconsider r as Real;
A12:  Partial_Sums(vp) is non-decreasing by A7,SERIES_1:16;
      now
        let n be set such that
A13:    n in dom Partial_Sums(vp);
        reconsider n1=n as Nat by A13;
        0<=vp.0 by A7;
        then
A14:    0<=Partial_Sums(vp).0 by SERIES_1:def 1;
        Partial_Sums(vp).0<=Partial_Sums(vp).n1 by A12,SEQM_3:11;
        hence (Partial_Sums(vp).n) to_power p1 < r to_power p1 by A11,A10,A13
,A14,Th1;
      end;
      then consider q be Real such that
A15:  for n be set st n in dom Partial_Sums(vp) holds (Partial_Sums(
      vp).n) to_power p1 <q;
      reconsider uq=u as Real_Sequence by FUNCT_2:66;
A17:  seq_id(v)+seq_id(u) =seq_id(v+u) by RSSPACE:2;
      seq_id(u) rto_power p is summable by A1,A4,Def2;
      then Partial_Sums(up) is bounded_above by A7,SERIES_1:17;
      then consider r1 be Real such that
A18:  for n be object st n in dom Partial_Sums(up) holds Partial_Sums(up
      ).n<r1 by SEQ_2:def 1;
      reconsider r1 as Real;
A19:  Partial_Sums(up) is non-decreasing by A7,SERIES_1:16;
      now
        let n be set such that
A20:    n in dom Partial_Sums(up);
        reconsider n1=n as Nat by A20;
        0<=up.0 by A7;
        then
A21:    0<=Partial_Sums(up).0 by SERIES_1:def 1;
        Partial_Sums(up).0<=Partial_Sums(up).n1 by A19,SEQM_3:11;
        hence (Partial_Sums(up).n) to_power p1 < r1 to_power p1 by A11,A18,A20
,A21,Th1;
      end;
      then consider q1 be Real such that
A22:  for n be set st n in dom Partial_Sums(up) holds (Partial_Sums(
      up).n) to_power p1 <q1;
      set g = q + q1;
A24:  p * (1/p) = (p*1)/p by XCMPLX_1:74
        .= 1 by A1,XCMPLX_1:60;
      now
        let n be Nat;
A25:   n in NAT by ORDINAL1:def 12;
A26:    now
          assume (Partial_Sums((vq + uq) rto_power p).n) > 0;
          hence
          (Partial_Sums((vq + uq) rto_power p).n) to_power (1/p) to_power
p = (Partial_Sums((vq + uq) rto_power p).n) to_power ((1/p)*p) by POWER:33
            .= (Partial_Sums((vq + uq) rto_power p).n) by A24,POWER:25;
        end;
        NAT=dom Partial_Sums(up) by SEQ_1:2;
        then
A27:    (Partial_Sums(up).n) to_power p1 <q1 by A22,A25;
        NAT=dom Partial_Sums(vp) by SEQ_1:2;
        then
A28:    (Partial_Sums(up).n) to_power (1/p) + (Partial_Sums(vp).n)
        to_power (1/p) < g by A15,A27,XREAL_1:8,A25;
        (Partial_Sums((vq + uq) rto_power p).n) to_power (1/p) <= (
Partial_Sums(up).n) to_power (1/p) + (Partial_Sums(vp).n) to_power (1/p)
by A1,Th2;
        then
A29:    (Partial_Sums((vq + uq) rto_power p).n) to_power (1/p) < g by A28,
XXREAL_0:2;
A30:    now
          assume
A31:      (Partial_Sums((vq + uq) rto_power p).n) = 0;
          hence
          (Partial_Sums((vq + uq) rto_power p).n) to_power (1/p) to_power
          p = 0 to_power p by A11,POWER:def 2
            .= (Partial_Sums((vq + uq) rto_power p).n) by A1,A31,POWER:def 2;
        end;
        thus
A32:    now
          let n be Nat;
          thus
A33:      0 < |.(vq + uq).n.| or 0=|.(vq + uq).n.| by COMPLEX1:46;
          ((vq + uq) rto_power p).n =|.(vq + uq).n.| to_power p by Def1;
          hence 0 <= ((vq + uq) rto_power p).n by A1,A33,POWER:34,def 2;
        end;
        then
A34:    0<=((vq + uq) rto_power p).0;
A35:    Partial_Sums((vq + uq) rto_power p).0 <=Partial_Sums((vq + uq)
        rto_power p).n by A32,SEQM_3:11,SERIES_1:16;
        then 0<=Partial_Sums((vq + uq) rto_power p).n by A34,SERIES_1:def 1;
        then
        (Partial_Sums((vq + uq) rto_power p).n) to_power (1/p) >=0 by A11,Lm1;
        hence
        (Partial_Sums((vq + uq) rto_power p).n) < g to_power p by A1,A29,A26
,A30,A35,A34,Th1,SERIES_1:def 1;
      end;
      then for n be Nat holds Partial_Sums((vq + uq) rto_power p)
      is bounded_above & 0 <= ((vq + uq) rto_power p).n by SEQ_2:def 3;
      hence thesis by A17,SERIES_1:17;
    end;
    thus thesis by A1,A5,Def2;
  end;
  for a be Real
  for v be VECTOR of Linear_Space_of_RealSequences st v in
  W holds a * v in W
  proof
    let a be Real;
    let v be VECTOR of Linear_Space_of_RealSequences such that
A37: v in W;
    seq_id(a*v) rto_power p is summable
    proof
      set vp = seq_id(v) rto_power p;
A38:  now
        let n be Nat;
        thus 0 <= |.(seq_id(v)).n.| by COMPLEX1:46;
        thus
A39:    0 < |.(seq_id(v)).n.| or 0=|.(seq_id(v)).n.| by COMPLEX1:46;
        vp.n=|.(seq_id(v)).n.| to_power p by Def1;
        hence 0 <= vp.n by A1,A39,POWER:34,def 2;
      end;
      vp is summable by A1,A37,Def2;
      then Partial_Sums(vp) is bounded_above by A38,SERIES_1:17;
      then consider r be Real such that
A40:  for n be object st n in dom Partial_Sums(seq_id(v) rto_power p)
      holds Partial_Sums(vp).n<r by SEQ_2:def 1;
A41:  seq_id(a*v)=(a(#)seq_id(v)) by RSSPACE:3;
A42:  for n be Nat holds Partial_Sums(seq_id(a*v) rto_power p)
      .n = (|.a.| to_power p)* Partial_Sums(seq_id(v) rto_power p).n
      proof
        let n be Nat;
        now
          let n be Nat;
A43:      |.a.|>=0 by COMPLEX1:46;
A44:      |.(seq_id(v)).n.| >= 0 by COMPLEX1:46;
A45:      (a(#)seq_id(v)).n = a * (seq_id(v)).n by SEQ_1:9;
          ((a(#)seq_id(v)) rto_power p).n = |.(a(#)seq_id(v)).n.|
          to_power p by Def1
            .=((|.a.|* |.(seq_id(v)).n.|)) to_power p by A45,COMPLEX1:65
            .=(|.a.| to_power p) * ((|.(seq_id(v)).n.|) to_power p) by A1,A43
,A44,Lm2;
          hence
          (seq_id(a*v) rto_power p).n = (|.a.| to_power p) * ((seq_id(v)
          ) rto_power p).n by A41,Def1
            .= ((|.a.| to_power p) (#) ((seq_id(v)) rto_power p)).n by SEQ_1:9;
        end;
        then
        for n be object st n in NAT holds (seq_id(a*v) rto_power p).n= ((
        |.a.| to_power p) (#) ((seq_id(v)) rto_power p)).n;
        then
A46:    (seq_id(a*v) rto_power p)= ((|.a.| to_power p) (#) ((seq_id(v))
        rto_power p)) by FUNCT_2:12;
        Partial_Sums(((|.a.| to_power p) (#) ((seq_id(v)) rto_power p)))
= ((|.a.| to_power p) (#)Partial_Sums( ((seq_id(v)) rto_power p))) by
SERIES_1:9;
        hence thesis by A46,SEQ_1:9;
      end;
A47:  0<(|.a.| to_power p) or 0=(|.a.| to_power p) by A1,Lm1,COMPLEX1:46;
A48:  now
        let n be set such that
A49:    n in dom Partial_Sums(seq_id(v) rto_power p);
        dom Partial_Sums(seq_id(v) rto_power p)= NAT by SEQ_1:1;
        hence n in dom Partial_Sums(seq_id(a*v) rto_power p) by A49,SEQ_1:1;
        thus (|.a.| to_power p) * Partial_Sums(seq_id(v) rto_power p).n < (
        |.a.| to_power p) * r or ((|.a.| to_power p) * Partial_Sums(seq_id(v)
        rto_power p).n ) = (|.a.| to_power p) * r by A40,A47,A49,XREAL_1:68;
      end;
A50:  for n be set st n in dom Partial_Sums(seq_id(a*v) rto_power p)
      holds Partial_Sums(seq_id(a*v) rto_power p).n <(|.a.| to_power p) * r or
      Partial_Sums(seq_id(a*v) rto_power p).n = (|.a.| to_power p) * r
      proof
        let n be set such that
A51:    n in dom Partial_Sums(seq_id(a*v) rto_power p);
        reconsider n1=n as Nat by A51;
        n in NAT by A51,SEQ_1:1;
        then n in dom Partial_Sums(seq_id(v) rto_power p) by SEQ_1:1;
        then (|.a.| to_power p) * Partial_Sums(seq_id(v) rto_power p). n < (
        |.a.| to_power p) * r or ((|.a.| to_power p) * Partial_Sums(seq_id(v)
        rto_power p).n ) = (|.a.| to_power p) * r by A48;
        then Partial_Sums(seq_id(a*v) rto_power p).n1 <(|.a.| to_power p) *
r or Partial_Sums(seq_id(a*v) rto_power p).n1 =(|.a.| to_power p) * r by A42;
        hence thesis;
      end;
      ex r1 be Real st for n be object st n in dom Partial_Sums(
      seq_id(a*v) rto_power p) holds Partial_Sums(seq_id(a*v) rto_power p).n<r1
      proof
        take r1 = (|.a.| to_power p) * r + 1;
        reconsider r1 as Real;
        for n be object st n in dom Partial_Sums(seq_id(a*v) rto_power p)
        holds Partial_Sums(seq_id(a*v) rto_power p).n<r1
        proof
A52:      (|.a.| to_power p) * r <(|.a.| to_power p) * r +1 by XREAL_1:29;
          let n be object;
          assume n in dom Partial_Sums(seq_id(a*v) rto_power p);
          hence thesis by A50,A52,XXREAL_0:2;
        end;
        hence thesis;
      end;
      then
A53:  Partial_Sums(seq_id(a*v) rto_power p) is bounded_above by SEQ_2:def 1;
      for n be Nat holds (seq_id(a*v) rto_power p).n >= 0
      proof
        set b = (a(#)seq_id(v));
        let n be Nat;
        (a(#)seq_id(v)).n = a * (seq_id(v)).n by SEQ_1:9;
        then (b rto_power p).n = |.a * (seq_id(v)).n.| to_power p by Def1;
        hence thesis by A1,A41,Lm1,COMPLEX1:46;
      end;
      hence thesis by A53,SERIES_1:17;
    end;
    hence thesis by A1,Def2;
  end;
  hence thesis by A2,RLSUB_1:def 1;
end;
