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 Th24:
   D.(Sum(LBZ(D,n+1,x,y))) = Sum(LBZ(D,n+2,x,y))
   proof
     set p1 = <*((D|^(n+1)).x)*y*>;
     set p2 = LBZ0(D,n,x,y);
     set p3 = <*x*((D|^(n+1)).y)*>;
     set q = LBZ(D,n+1,x,y);
A1:  D.(((D|^(n+1)).x)*y) = D.((D|^(n+1)).x)*y +((D|^(n+1)).x)*D.y by Def1
     .= ((D|^(n+1+1)).x)*y + ((D|^(n+1)).x)*D.y by Th9;
A2:  D.(x*((D|^(n+1)).y)) = (D.x)*((D|^(n+1)).y) + x*D.((D|^(n+1)).y) by Def1
     .= (D.x)*((D|^(n+1)).y) + x*((D|^(n+1+1)).y) by Th9;
A3:  Sum(D*p2)
      = -(LBZ1(D,n+1,x,y)/.1)+Sum LBZ0(D,n+1,x,y)-(LBZ2(D,n+1,x,y)/.(n+1))
        by Th21
     .= -(((D|^(n+1)).x)*D.y) +Sum LBZ0(D,n+1,x,y)-(LBZ2(D,n+1,x,y)/.(n+1))
        by NAT_1:12,Lm1
     .= -(((D|^(n+1)).x)*D.y) +Sum LBZ0(D,n+1,x,y)-((D.x)*((D|^(n+1)).y))
        by NAT_1:12,Lm2;
     D.(Sum(LBZ(D,n+1,x,y))) = Sum(D*(LBZ(D,n+1,x,y))) by Th11
     .= Sum(D*(p1^p2^p3)) by Th22
     .= Sum( (D*(p1^p2))^(D*p3)) by FINSEQOP:9
     .= Sum(D*(p1^p2))+ Sum(D*p3) by RLVECT_1:41
     .= Sum((D*p1)^(D*p2)) + Sum(D*p3) by FINSEQOP:9
     .= Sum(D*p1)+ Sum(D*p2)+ Sum(D*p3) by RLVECT_1:41
     .= Sum(<*D.(((D|^(n+1)).x)*y)*>)+ Sum(D*p2)+ Sum(D*p3) by FINSEQ_2:35
     .= D.(((D|^(n+1)).x)*y)+ Sum(D*p2)+ Sum(D*p3) by BINOM:3
     .= D.(((D|^(n+1)).x)*y)+ Sum(D*p2)+ Sum(<* D.(x*((D|^(n+1)).y))*>)
        by FINSEQ_2:35
     .= ((D|^(n+1+1)).x)*y + ((D|^(n+1)).x)*D.y
        + (-((D|^(n+1)).x)*D.y + Sum(LBZ0(D,n+1,x,y))-(D.x)*((D|^(n+1)).y))
        + D.(x*((D|^(n+1)).y)) by A3,A1,BINOM:3
     .= ((D|^(n+1+1)).x)*y + ((D|^(n+1)).x)*D.y
        + (-((D|^(n+1)).x)*D.y +(Sum( LBZ0(D,n+1,x,y)) -(D.x)*((D|^(n+1)).y)))
        + D.(x*((D|^(n+1)).y)) by RLVECT_1:def 3
     .= ((D|^(n+1+1)).x)*y + ((D|^(n+1)).x)*D.y
        -((D|^(n+1)).x)*D.y +((Sum( LBZ0(D,n+1,x,y)) -(D.x)*((D|^(n+1)).y)))
        + D.(x*((D|^(n+1)).y)) by RLVECT_1:def 3
     .= ((D|^(n+1+1)).x)*y + (((D|^(n+1)).x)*D.y
        -((D|^(n+1)).x)*D.y) +((Sum( LBZ0(D,n+1,x,y)) -(D.x)*((D|^(n+1)).y)))
        + D.(x*((D|^(n+1)).y)) by RLVECT_1:def 3
     .= ((D|^(n+1+1)).x)*y + (0.R)
        +((Sum( LBZ0(D,n+1,x,y)) -(D.x)*((D|^(n+1)).y)))
        + D.(x*((D|^(n+1)).y)) by RLVECT_1:15
     .= ((D|^(n+1+1)).x)*y
        + Sum( LBZ0(D,n+1,x,y))+(-(D.x)*((D|^(n+1)).y))
        + ((D.x)*((D|^(n+1)).y) + x*((D|^(n+1+1)).y)) by A2,RLVECT_1:def 3
     .= ((D|^(n+1+1)).x)*y
        + Sum( LBZ0(D,n+1,x,y))+((-(D.x)*((D|^(n+1)).y))
        + ((D.x)*((D|^(n+1)).y) + x*((D|^(n+1+1)).y))) by RLVECT_1:def 3
     .= ((D|^(n+1+1)).x)*y
        + Sum( LBZ0(D,n+1,x,y))+(( -(D.x)*((D|^(n+1)).y)
        + (D.x)*((D|^(n+1)).y)) + x*((D|^(n+1+1)).y)) by RLVECT_1:def 3
     .= ((D|^(n+1+1)).x)*y
        + Sum( LBZ0(D,n+1,x,y))+((0.R) + x*((D|^(n+1+1)).y)) by RLVECT_1:5
     .= Sum(<*((D|^(n+1+1)).x)*y*>^(LBZ0(D,n+1,x,y))^<*x*((D|^(n+1+1)).y)*>)
        by Th23
     .= Sum LBZ(D,n+1+1,x,y) by Th22;
     hence thesis;
   end;
