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
  seq1 (##) (seq2 + seq3) = (seq1 (##) seq2) + (seq1 (##) seq3)
proof
  set S=seq2 + seq3;
  set S2=seq1 (##) seq2;
  set S3=seq1 (##) seq3;
  now
    let x be object;
    assume x in NAT;
    then reconsider k=x as Nat;
    consider Fr be XFinSequence of REAL such that
A1: dom Fr = k+1 and
A2: for n st n in k+1 holds Fr.n = seq1.n * S.(k-'n) and
A3: Sum Fr = (seq1(##)S).k by Def4;
    consider Fr1 be XFinSequence of REAL such that
A4: dom Fr1 = k+1 and
A5: for n st n in k+1 holds Fr1.n = seq1.n * seq2.(k-'n) and
A6: Sum Fr1 = S2.k by Def4;
A7: len Fr1=len Fr by A1,A4;
    consider Fr2 be XFinSequence of REAL such that
A8: dom Fr2 = k+1 and
A9: for n st n in k+1 holds Fr2.n = seq1.n * seq3.(k-'n) and
A10: Sum Fr2 = S3.k by Def4;
A11: for n be Nat st n in dom Fr holds Fr.n=addreal.(Fr1.n,Fr2.n)
    proof
      let n be Nat such that
A12:  n in dom Fr;
A13:  Fr.n = seq1.n * S.(k-'n) by A1,A2,A12;
A14:  S.(k-'n)=seq2.(k-'n) + seq3.(k-'n) by SEQ_1:7;
      Fr1.n = seq1.n * seq2.(k-'n) & Fr2.n = seq1.n * seq3.(k-'n) by A1,A5,A9
,A12;
      then Fr.n=Fr1.n+Fr2.n by A13,A14;
      hence thesis by BINOP_2:def 9;
    end;
    len Fr1=len Fr2 by A4,A8;
  then addreal "**" Fr1^Fr2 = addreal "**" Fr by A11,A7,AFINSQ_2:46;
   then Sum Fr=addreal "**" Fr1^Fr2 by AFINSQ_2:48;
  then Sum Fr=Sum (Fr1^Fr2) by AFINSQ_2:48;
then Sum Fr=Sum (Fr1)+Sum(Fr2) by AFINSQ_2:55;
    hence (seq1(##)S).x=(S2+S3).x by A3,A6,A10,SEQ_1:7;
  end;
  hence thesis by FUNCT_2:12;
end;
