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 Th16:
       LBZ(D,1,x,y) = <* y*D.x, x*D.y *>
       proof
         len LBZ(D,1,x,y) = 1+1 by Def4; then
         dom(LBZ(D,1,x,y)) = Seg 2 by FINSEQ_1:def 3; then
A2:      1 in dom(LBZ(D,1,x,y)) & 2 in dom(LBZ(D,1,x,y));
A3:      1 choose (1-'1) = 1 choose 0 by XREAL_1:232 .= 1 by NEWTON:19;
A4:      2-'1 = 2-1 by XREAL_1:233; then
A5:      1 choose (2-'1) = 1 by NEWTON:21;
A6:      LBZ(D,1,x,y).1 = 1*((D|^(1+1-'1)).x)*((D|^(1 -' 1)).y) by A3,A2,Def4
         .= ((D|^(2-'1)).x)*((D|^(1 -' 1)).y) by BINOM:13
         .= ((D|^1).x)*((D|^0).y) by XREAL_1:232,A4
         .= ((D|^1).x)*((id R).y) by VECTSP11:18
         .= y*D.x by VECTSP11:19;
         LBZ(D,1,x,y).2 = 1*((D|^(1+1-'2)).x)*((D|^(2 -' 1)).y) by A5,A2,Def4
         .= ((D|^(2-'2)).x)*((D|^(2 -' 1)).y) by BINOM:13
         .= ((D|^0).x)*((D|^1).y) by XREAL_1:232,A4
         .= ((id R).x)*((D|^1).y) by VECTSP11:18
         .= x*D.y by VECTSP11:19;
         hence thesis by Def4,A6,FINSEQ_1:44;
       end;
