reserve L for Abelian left_zeroed add-associative associative right_zeroed
              right_complementable distributive non empty doubleLoopStr;
reserve a,b,c for Element of L;
reserve R for non degenerated comRing;
reserve n,m,i,j,k for Nat;
 reserve D for Function of R, R;
 reserve x,y,z for Element of R;
reserve D for Derivation of R;
reserve s for FinSequence of the carrier of R;
reserve h for Function of R,R;

theorem Th18:
   LBZ0(D,m,x,y) = LBZ1(D,m,x,y) + LBZ2(D,m,x,y)
   proof
    set p = LBZ1(D,m,x,y);
    set q = LBZ2(D,m,x,y);
    set r = LBZ0(D,m,x,y);
A1: dom p = Seg len p by FINSEQ_1:def 3 .= Seg m by Def6;
A2: dom q = Seg len q by FINSEQ_1:def 3 .= Seg m by Def7;
A3: dom r = Seg len r by FINSEQ_1:def 3 .= Seg m by Def5;
A4: dom (p + q) = Seg m by A1,BINOM:def 1;
A5: Seg len (p+q) = dom (p + q) by FINSEQ_1:def 3 .= dom p by BINOM:def 1
    .= Seg len p by FINSEQ_1:def 3 .= Seg m by Def6;
A6: len r = m by Def5 .= len (p+q) by A5,FINSEQ_1:6;
    for k st 1 <= k & k <= len (p+q) holds (p+q).k = r.k
    proof
      let k be Nat;
      assume
A7:   1 <= k & k <= len (p+q); then
A8:   k in dom p by A1,A5; then
A9:   p/.k = p.k by PARTFUN1:def 6
      .= (m choose (k-'1))*((D|^(m+1 -'k)).x)*((D|^k).y) by A8,Def6;
A10:  k in dom q by A2,A5,A7; then
A11:  q/.k = q.k by PARTFUN1:def 6
      .= (m choose k)*((D|^(m +1 -'k)).x)*((D|^k).y) by A10,Def7;
A12:  k in dom r by A3,A5,A7;
      k in dom (p+q) by A4,A5,A7; then
      (p+q).k = (p+q)/.k by PARTFUN1:def 6
      .= (m choose (k-'1))*((D|^(m+1 -'k)).x)*((D|^k).y)
       + (m choose k)*((D|^(m+1 -'k)).x)*((D|^k).y) by A11,A9,A7,BINOM:def 1
      .= (m choose (k-'1))*(((D|^(m+1 -'k)).x)*((D|^k).y))
       + (m choose k)*((D|^(m+1 -'k)).x)*((D|^k).y) by BINOM:19
      .= (m choose (k-'1))*(((D|^(m+1 -'k)).x)*((D|^k).y))
       + (m choose k)*(((D|^(m+1 -'k)).x)*((D|^k).y)) by BINOM:19
      .= ((m choose (k-'1))+(m choose k))*(((D|^(m+1 -'k)).x)*((D|^k).y))
          by BINOM:15
      .= ((m choose (k-'1))+(m choose k))*((D|^(m+1 -'k)).x)*((D|^k).y)
         by BINOM:19
      .= r.k by A12,Def5;
      hence thesis;
    end;
    hence thesis by A6;
  end;
