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 Th12:
  for K being add-associative right_zeroed right_complementable
     non empty addLoopStr
  for p being FinSequence of K for i st i in dom p &
  for k st k in dom p & k<>i holds p.k=0.K holds Sum p=p.i
proof
  let K be add-associative right_zeroed right_complementable non empty
  addLoopStr;
  let p be FinSequence of K;
  let i;
  assume that
A1: i in dom p and
A2: for k st k in dom p & k<>i holds p.k=0.K;
  reconsider a=p.i as Element of K by A1,FINSEQ_2:11;
  reconsider p1=p|Seg i as FinSequence of K by FINSEQ_1:18;
  i<>0 by A1,FINSEQ_3:25;
  then i in Seg i by FINSEQ_1:3;
  then i in (dom p) /\ (Seg i) by A1,XBOOLE_0:def 4;
  then
A3: i in dom p1 by RELAT_1:61;
  then p1 <> {};
  then len p1<>0;
  then consider p3 being FinSequence of K, x being Element of K such that
A4: p1=p3^<*x*> by FINSEQ_2:19;
  p1 is_a_prefix_of p by TREES_1:def 1;
  then consider p2 being FinSequence such that
A5: p=p1^p2 by TREES_1:1;
  reconsider p2 as FinSequence of K by A5,FINSEQ_1:36;
A6: dom p2 = Seg len p2 by FINSEQ_1:def 3;
A7: for k st k in Seg len p2 holds p2.k=0.K
  proof
    let k;
A8: i <=len p1 & len p1 <= len p1 + k by A3,FINSEQ_3:25,NAT_1:12;
    assume k in Seg len p2;
    then
A9: k in dom p2 by FINSEQ_1:def 3;
    then 0<>k by FINSEQ_3:25;
    then
A10: i<>len p1 + k by A8,XCMPLX_1:3,XXREAL_0:1;
    thus p2.k=p.(len p1+k) by A5,A9,FINSEQ_1:def 7
      .=0.K by A2,A5,A9,A10,FINSEQ_1:28;
  end;
A11: now
    let j be Nat;
    assume
A12: j in dom p2;
    hence p2.j =0.K by A7,A6
      .= (len p2 |->0.K).j by A6,A12,FINSEQ_2:57;
  end;
A13: dom p3 = Seg len p3 by FINSEQ_1:def 3;
  i <= len p by A1,FINSEQ_3:25;
  then
A14: i =len p1 by FINSEQ_1:17
    .=len p3+ len <*x*> by A4,FINSEQ_1:22
    .= len p3 + 1 by FINSEQ_1:39;
  then
A15: x =p1.i by A4,FINSEQ_1:42
    .=a by A5,A3,FINSEQ_1:def 7;
A16: for k st k in Seg len p3 holds p3.k=0.K
  proof
    let k;
    assume
A17: k in Seg len p3;
    then k <= len p3 by FINSEQ_1:1;
    then
A18: i<> k by A14,NAT_1:13;
A19: k in dom p3 by A17,FINSEQ_1:def 3;
    then
A20: k in dom p1 by A4,FINSEQ_2:15;
    thus p3.k=p1.k by A4,A19,FINSEQ_1:def 7
      .=p.k by A5,A20,FINSEQ_1:def 7
      .=0.K by A2,A5,A18,A20,FINSEQ_2:15;
  end;
A21: now
    let j be Nat;
    assume
A22: j in dom p3;
    hence p3.j = 0.K by A16,A13
      .= (len p3 |->0.K).j by A13,A22,FINSEQ_2:57;
  end;
  len (len p3 |->0.K)=len p3 by CARD_1:def 7;
  then
A23: p3=len p3 |->0.K by A21,FINSEQ_2:9;
  len (len p2 |->0.K)=len p2 by CARD_1:def 7;
  then p2=len p2 |->0.K by A11,FINSEQ_2:9;
  then Sum p=Sum(p3^<*x*>) + Sum(len p2 |->0.K) by A5,A4,RLVECT_1:41
    .=Sum(p3^<*x*>) + 0.K by Th11
    .=Sum(p3^<*x*>) by RLVECT_1:4
    .=Sum(len p3 |->0.K) + x by A23,FVSUM_1:71
    .=0.K + a by A15,Th11
    .=p.i by RLVECT_1:4;
  hence thesis;
end;
