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 Th13:
    for s holds len s = 1 implies Sum DProd(D,s)= D.(Product s)
    proof
      let s;
      assume
A1:   len s = 1;
      reconsider Ds1 = D.(s/.1) as Element of R;
      dom s = Seg 1 by A1,FINSEQ_1:def 3; then
A2:   1 in dom s;
A3:   s = <* s.1 *> by A1, FINSEQ_1:40 .= <* s/.1 *> by A2,PARTFUN1:def 6;
      reconsider s1 = s/.1 as Element of R;
A4:   len DProd(D,s) = 1 by A1,Def3;
      dom DProd(D,s) = Seg 1 by A4,FINSEQ_1:def 3; then
A5:   1 in dom DProd(D,s);
A6:   Replace(s,1,Ds1) = <* Ds1 *> by A3,FINSEQ_7:12;
      Sum DProd(D,s) = DProd(D,s).1 by A4, RLVECT_1:76
      .= Product <* D.(s/.1) *> by A6,A5,Def3
      .= D.(s/.1) by GROUP_4:9
      .= D.(Product s) by A3,GROUP_4:9;
      hence thesis;
    end;
