reserve x,y,z,x1,x2,y1,y2,z1,z2 for object,
  i,j,k,l,n,m for Nat,
  D for non empty set,
  K for Ring;

theorem Th17:
  for K being Ring
  for p,q being FinSequence of K for i st i in dom p & i in dom q
  & p.i=1.K & for k st k in dom p & k<>i holds p.k=0.K holds
    Sum(mlt(p,q))=q.i & Sum(mlt(q,p)) = q.i
proof
  let K be Ring;
  let p,q be FinSequence of K;
  let i;
  assume that
A1: i in dom p & i in dom q and
A2: p.i=1.K & for k st k in dom p & k<>i holds p.k=0.K;
  thus Sum(mlt(p,q))=q.i
  proof
    reconsider r=mlt(p,q) as FinSequence of K;
A3: for k st k in dom r & k<>i holds r.k=0.K by A2,Th14;
A4: dom p = Seg len p & dom q = Seg len q by FINSEQ_1:def 3;
    dom (mlt(p,q)) = Seg len (mlt(p,q)) & len (mlt(p,q))=min(len p,len q)
      by Th13,FINSEQ_1:def 3;
    then dom p /\ dom q = dom (mlt(p,q)) by A4,FINSEQ_2:2;
    then
A5: i in dom r by A1,XBOOLE_0:def 4;
    then r.i=q.i by A2,Th14;
    hence thesis by A5,A3,Th12;
  end;
  thus Sum(mlt(q,p))=q.i
  proof
    reconsider r=mlt(q,p) as FinSequence of K;
A6: for k st k in dom r & k<>i holds r.k=0.K by A2,Th14;
A7: dom p = Seg len p & dom q = Seg len q by FINSEQ_1:def 3;
    dom (mlt(q,p)) = Seg len (mlt(q,p)) & len (mlt(q,p))=min(len q,len p)
      by Th13,FINSEQ_1:def 3;
    then dom q /\ dom p = dom (mlt(q,p)) by A7,FINSEQ_2:2;
    then
A8: i in dom r by A1,XBOOLE_0:def 4;
    then r.i=q.i by A2,Th14;
    hence thesis by A8,A6,Th12;
  end;
end;
