reserve x, x1, x2, y, X, D for set,
  i, j, k, l, m, n, N for Nat,
  p, q for XFinSequence of NAT,
  q9 for XFinSequence,
  pd, qd for XFinSequence of D;
reserve pN, qN for Element of NAT^omega;
reserve seq1,seq2,seq3,seq4 for Real_Sequence,
  r,s,e for Real,
  Fr,Fr1, Fr2 for XFinSequence of REAL;

theorem Th53:
  for seq1, seq2 st seq1 is absolutely_summable & seq2 is summable
  holds seq1 (##) seq2 is summable & Sum(seq1 (##) seq2) = Sum seq1 * Sum seq2
proof
  let seq1, seq2 such that
A1: seq1 is absolutely_summable and
A2: seq2 is summable;
  set S2=Sum seq2;
  set S1=Sum seq1;
  set PA=Partial_Sums(|.seq1.|);
  set P2=Partial_Sums(seq2);
  set P1=Partial_Sums(seq1);
  set S=seq1 (##) seq2;
  set P=Partial_Sums(S);
A3: for e st 0<e ex n be Nat st
   for m be Nat st n<=m holds |.P.m-S1*S2.|<e
  proof
    seq1 is summable by A1;
    then
A4: P1 is convergent by SERIES_1:def 2;
    let e such that
A5: 0<e;
    set e1=e/(3*(|.S2.|+1));
    (|.S2.|+1)> (0 qua Nat)+(0 qua Nat) by COMPLEX1:46,XREAL_1:8;
    then
A6: 3*(|.S2.|+1)>3*0 by XREAL_1:68;
    then lim P1= S1 & e1>0 by A5,SERIES_1:def 3,XREAL_1:139;
    then consider n0 be Nat such that
A7: for n be Nat st n0 <= n holds |.P1.n-S1.| < e1
      by A4,SEQ_2:def 7;
    set e3=e/(3*(Sum abs(seq1)+1));
    |.S2.|+1>(0 qua Nat)+(0 qua Nat) by COMPLEX1:46,XREAL_1:8;
    then
A8: e1*(|.S2.|+1)=e/3 by XCMPLX_1:92;
A9: P2 is convergent & lim P2= S2 by A2,SERIES_1:def 2,def 3;
    consider r such that
A10: 0 < r and
A11: for k holds |.Sum(seq2^\k).| < r by A2,Th51;
    set e2=e/(3*r);
A12: |.S2.|+1 > |.S2.|+(0 qua Nat) by XREAL_1:8;
A13: now
      let n;
      |.seq1.n.|=abs(seq1).n by SEQ_1:12;
      hence abs(seq1).n>=0 by COMPLEX1:46;
    end;
then A14:for n be Nat holds |.seq1.|.n>=0;
A15: abs seq1 is summable by A1,SERIES_1:def 4;
    then Sum abs(seq1)>=0 by A14,SERIES_1:18;
    then
A16: (Sum abs(seq1)+1)*e3=e/3 by XCMPLX_1:92;
A17: Sum abs seq1>=0 by A15,A14,SERIES_1:18;
    then 3*(Sum abs(seq1)+1)>0*3 by XREAL_1:68;
    then e3>0 by A5,XREAL_1:139;
    then consider n2 be Nat such that
A18: for n be Nat st n2<=n holds |.P2.n-S2.|<e3 by A9,SEQ_2:def 7;
    3*r>0*3 by A10,XREAL_1:68;
    then e2>0 by A5,XREAL_1:139;
    then consider n1 be Nat such that
A19: for n be Nat
     st n1 <= n holds |.PA.n-PA.n1.| < e2 by A15,SERIES_1:21;
    reconsider M=max(max(1,n0),max(n1+1,n2)) as Nat by TARSKI:1;
A20: max(n1+1,n2)<=M by XXREAL_0:25;
    take 2M=M*2;
    let m be Nat such that
A21: 2M <= m;
A22: max(1,n0)<=M by XXREAL_0:25;
    then 0<M by XXREAL_0:25;
    then reconsider M1=M-1 as Nat by NAT_1:20;
A23: M=M1+1;
A24: n1+1 <= max(n1+1,n2) by XXREAL_0:25;
    then M1+1>=n1+1 by A20,XXREAL_0:2;
    then M1>= n1 by XREAL_1:8;
    then PA.M1>=PA.n1 by A13,Th52;
    then PA.m-PA.M1<=PA.m-PA.n1 by XREAL_1:10;
    then
A25: r*(PA.m-PA.M1)<=r*(PA.m-PA.n1) by A10,XREAL_1:64;
    consider Fr such that
A26: P.m= S2 * P1.m- Sum Fr and
A27: dom Fr=m+1 and
A28: for i st i in m+1 holds Fr.i=seq1.i*Sum(seq2^\(m-'i+1))by A2,Th49;
    consider absFr be XFinSequence of REAL such that
A29: dom absFr=dom Fr and
A30: |.Sum Fr.| <= Sum absFr and
A31: for i st i in dom absFr holds absFr.i=|.Fr.i.| by Th50;
A32: M<=M+M by NAT_1:11;
    then
A33: M<=m by A21,XXREAL_0:2;
    then M < len absFr by A27,A29,NAT_1:13;
    then
A34: len (absFr|M)=M by AFINSQ_1:11;
    n1+1 <= M by A24,A20,XXREAL_0:2;
    then n1+1<=m by A33,XXREAL_0:2;
    then
A35: n1 <= m by NAT_1:13;
    then PA.m>=PA.n1 by A13,Th52;
    then PA.m-PA.n1>=PA.n1-PA.n1 by XREAL_1:9;
    then
A36: |.PA.m-PA.n1.|=(PA.m-PA.n1) by ABSVALUE:def 1;
    consider Fr1 such that
A37: absFr=absFr|M^Fr1 by Th1;
A38: m+1=len (absFr|M)+ len Fr1 by A27,A29,A37,AFINSQ_1:def 3;
    then
A39: Fr1|((m-M)+1)=Fr1 by A34;
A40: n2 <= max(n1+1,n2) by XXREAL_0:25;
    then n2 <= M by A20,XXREAL_0:2;
    then n2 <=2M by A32,XXREAL_0:2;
    then n2 <= m & m in NAT by A21,XXREAL_0:2, ORDINAL1:def 12;
    then
A41: |.P2.m-S2.|<e3 by A18;
    defpred S[Nat] means M+$1+1<=m+1 implies Sum(Fr1|($1+1))<=r*(PA
    .(M+$1)-PA.M1);
A42: for k st S[k] holds S[k+1]
    proof
      let k such that
A43:  S[k];
      set k1=k+1;
      set Mk1=M+k1;
      reconsider Mk=M+k as Nat;
A44:  |.seq1.Mk1.|=abs(seq1).Mk1 by SEQ_1:12;
      assume
A45:  Mk1+1<=m+1;
      then
 A46:  Mk1 <m+1 by NAT_1:13;
      then
A47:  Mk1 in Segm(m+1) by NAT_1:44;
      then Fr.Mk1=seq1.Mk1*Sum(seq2^\(m-'Mk1+1)) by A28;
      then
A48:  |.Fr.Mk1.|=|.seq1.Mk1.|*|.Sum(seq2^\(m-'Mk1+1)).| by COMPLEX1:65;
      Mk1<m+1 by A45,NAT_1:13;
      then k1 < len Fr1 by A38,A34,XREAL_1:7;
      then k1 in len Fr1 by AFINSQ_1:86;
      then
A49:  Sum(Fr1|(k1+1))=Fr1.k1+Sum(Fr1|k1) by AFINSQ_2:65;
      m+1=len absFr by A27,A29;
      then absFr.Mk1=Fr1.(Mk1-M) by A37,A34,A46,AFINSQ_1:19,NAT_1:11;
      then
A50:  Fr1.k1=|.Fr.Mk1.| by A27,A29,A31,A47;
A51:   (|.seq1.|.(Mk+1)+PA.(Mk)) = PA.(Mk+1) by SERIES_1:def 1;
      |.seq1.Mk1.|>=0 & |.Sum(seq2^\(m-'Mk1+1)).|<r by A11,COMPLEX1:46;
      then Fr1.k1<=r*|.seq1.Mk1.| by A50,A48,XREAL_1:64;
      then Sum(Fr1|(k1+1))<=r*|.seq1.|.Mk1+r*(PA.(M+k)-PA.M1)by A43,A45,A49
,A44,NAT_1:13,XREAL_1:7;
      then Sum(Fr1|(k1+1))<=r*((|.seq1.|.Mk1+PA.(M+k))-PA.M1);
      then Sum(Fr1|(k1+1))<=r*(PA.(M+k+1)-PA.M1) by A51;
      hence thesis;
    end;
    |.PA.m - PA.n1.| < e2 by A19,A35;
    then r*(PA.m-PA.n1)<=r*e2 by A10,A36,XREAL_1:64;
    then
A52: r*(PA.m-PA.M1)<=r*e2 by A25,XXREAL_0:2;
A53: m=M+(m-M) & m-M=m-'M by A21,A32,XREAL_1:233,XXREAL_0:2;
A54: S[0]
    proof
      assume
A55:  M+(0 qua Nat)+1<=m+1;
      then
A56:  M <m+1 by NAT_1:13;
      then
A57:  M in Segm(m+1) by NAT_1:44;
      then
A58:  Fr.M=seq1.M*Sum(seq2^\(m-'M+1)) by A28;
      M+1-M<=m+1-M by A55,XREAL_1:9;
      then Segm 1 c= Segm len Fr1 by A38,A34,NAT_1:39;
      then
A59:  dom (Fr1|1)=1 by RELAT_1:62;
      m+1=len absFr by A27,A29;
      then absFr.M=Fr1.(M-M) by A37,A34,A56,AFINSQ_1:19;
      then Fr1.0=|.Fr.M.| by A27,A29,A31,A57;
      then
A60:  Fr1.0=|.seq1.M.|*|.Sum(seq2^\(m-'M+1)).| by A58,COMPLEX1:65;
A61:  |.seq1.M.|>=0 & r>|.Sum(seq2^\(m-'M+1)) .| by A11,COMPLEX1:46;
      0 in Segm 1 by NAT_1:44;
      then
A62:  (Fr1|1).0=Fr1.0 by A59,FUNCT_1:47;
      PA.M1+|.seq1.|.(M1+1)=PA.(M1+1) by SERIES_1:def 1;
      then
A63:  PA.M-PA.M1=|.seq1.M.| by SEQ_1:12;
      Sum (Fr1|1)=(Fr1|1).0 by A59,Lm3;
      hence thesis by A62,A60,A63,A61,XREAL_1:64;
    end;
    for k holds S[k] from NAT_1:sch 2(A54,A42);
    then Sum(Fr1)<=r*(PA.m-PA.M1) by A39,A53;
    then Sum(Fr1)<=r*e2 by A52,XXREAL_0:2;
    then
A64: Sum(Fr1)<=e/3 by A10,XCMPLX_1:92;
    |.seq1.|.0>=0 by A13;
    then
A65: |.seq1.|.0*|.P2.m-S2.|<=e3*|.seq1.|.0 by A41,XREAL_1:64;
A66: 0 in Segm(m+1) by NAT_1:44;
    then
A67: Fr.0=seq1.0*Sum(seq2^\(m-'0+1)) by A28;
    PA.M1 <= Sum |.seq1.| by A14,A15,RSSPACE2:3;
    then
A68: e3* PA.M1 <= e3* Sum |.seq1.| by A5,A17,XREAL_1:64;
    S2=P2.(m-'0)+Sum(seq2^\(m-'0+1)) & m-'0=m by A2,NAT_D:40,SERIES_1:15;
    then
A69: Sum(seq2^\(m-'0+1))=S2-P2.m;
    n0<= max(1,n0) by XXREAL_0:25;
    then n0 <= M by A22,XXREAL_0:2;
    then n0 <= m & m in NAT by A33,XXREAL_0:2, ORDINAL1:def 12;
    then
A70: |.P1.m-S1.|<e1 by A7;
    |.S2*(P1.m-S1).|=|.S2.|*|.P1.m-S1.| & |.S2.|>=0 by COMPLEX1:46,65;
    then
A71: |.S2*(P1.m-S1).|<=|.S2.|*e1 by A70,XREAL_1:64;
A72: Sum absFr=Sum(absFr|M)+Sum Fr1 by A37,AFINSQ_2:55;
    defpred Q[Nat] means $1+1 <= M implies Sum(absFr|($1+1))<= e3 *
    PA.$1;
A73: n2 <= M by A40,A20,XXREAL_0:2;
A74: for k st Q[k] holds Q[k+1]
    proof
      let k such that
A75:  Q[k];
      reconsider k1=k+1 as Nat;
A76:  |.seq1.k1.|=abs(seq1).k1 by SEQ_1:12;
A77:  m-M >= 2M-M by A21,XREAL_1:9;
      assume
A78:  k+1+1<=M;
      then
A79:  k1<M by NAT_1:13;
      then m-k1 >= m-M by XREAL_1:10;
      then m-k1 >= M by A77,XXREAL_0:2;
      then
A80:  m-k1 >= n2 by A73,XXREAL_0:2;
      e3*|.seq1.k1.|+Sum(absFr|k1) <= e3*|.seq1.k1.| + e3 * PA.k by A75,A78,
NAT_1:13,XREAL_1:6;
      then e3*|.seq1.k1.|+Sum(absFr|k1)<=e3*(abs(seq1).k1+PA.k) & k in NAT
        by A76,ORDINAL1:def 12;
      then
A81:  e3*|.seq1.k1.|+Sum(absFr|k1)<= e3* PA.k1 by SERIES_1:def 1;
      k1<m by A33,A79,XXREAL_0:2;
      then k1 < m+1 by NAT_1:13;
      then
A82:  k1 in Segm(m+1) by NAT_1:44;
      then
A83:  Sum(absFr|(k1+1))=absFr.k1 + Sum(absFr|k1) by A27,A29,AFINSQ_2:65;
      m-k1=m-'k1 by A33,A79,XREAL_1:233,XXREAL_0:2;
      then |.P2.(m-'k1)-S2.|<e3 by A18,A80;
      then
A84:  |.S2-P2.(m-'k1).|<e3 by COMPLEX1:60;
A85:  S2=P2.(m-'k1)+Sum(seq2^\(m-'k1+1)) by A2,SERIES_1:15;
      |.seq1.k1.|>=0 by COMPLEX1:46;
      then |.S2-P2.(m-'k1).|*|.seq1.k1.| <= e3*|.seq1.k1.| by A84,XREAL_1:64;
      then
A86:
|.seq1.k1 * Sum(seq2^\(m-'k1+1)).| <= e3 * |.seq1.k1.| by A85,COMPLEX1:65;
      Fr.k1=seq1.k1*Sum(seq2^\(m-'k1+1)) & |.Fr.k1.|=absFr.k1 by A27,A28,A29
,A31,A82;
      then Sum(absFr|(k1+1))<=e3*|.seq1.k1.|+Sum(absFr|k1) by A83,A86,
XREAL_1:6;
      hence thesis by A81,XXREAL_0:2;
    end;
A87: Sum(absFr|zz) = 0;
    Sum(absFr|(zz+1))=absFr.0 + Sum(absFr|zz) & absFr.0=|.Fr.0.|
    by A27,A29,A31,A66,AFINSQ_2:65;
    then Sum(absFr|(zz+1))=|.seq1.0 .|*|.Sum(seq2^\(m-'0+1)).| by A67,A87,
COMPLEX1:65
      .=abs(seq1).0*|.Sum(seq2^\(m-'0+1)).| by SEQ_1:12
      .=abs(seq1).0*|.P2.m-S2.| by A69,COMPLEX1:60;
    then
A88: Q[0] by A65,SERIES_1:def 1;
    for k holds Q[k] from NAT_1:sch 2(A88,A74);
    then
A89: Sum(absFr|M)<= e3 * PA.M1 by A23;
    Sum abs(seq1)+1>Sum abs(seq1)+(0 qua Nat) by XREAL_1:8;
    then e3*(Sum abs(seq1)+1) >= e3*Sum abs(seq1) by A5,A17,XREAL_1:64;
    then e3* PA.M1 <=e/3 by A68,A16,XXREAL_0:2;
    then Sum(absFr|M)<=e/3 by A89,XXREAL_0:2;
    then Sum(absFr|M)+Sum Fr1 <=e/3+e/3 by A64,XREAL_1:7;
    then
A90: |.Sum Fr.|<= e/3+e/3 by A30,A72,XXREAL_0:2;
    P.m-S1*S2=S2*(P1.m-S1)-Sum Fr by A26;
    then
A91: |.P.m-S1*S2.|<=|.S2*(P1.m-S1).|+|.Sum Fr.| by COMPLEX1:57;
    e1>0 by A5,A6,XREAL_1:139;
    then e1*(|.S2.|+1) > e1*|.S2.| by A12,XREAL_1:68;
    then |.S2*(P1.m-S1).|<e/3 by A8,A71,XXREAL_0:2;
    then |.S2*(P1.m-S1).|+|.Sum Fr.| <e/3+(e/3+e/3) by A90,XREAL_1:8;
    hence thesis by A91,XXREAL_0:2;
  end;
  then
A92: P is convergent by SEQ_2:def 6;
  hence S is summable by SERIES_1:def 2;
  lim P = S1*S2 by A3,A92,SEQ_2:def 7;
  hence thesis by SERIES_1:def 3;
end;
