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 Th15:
       LBZ(D,0,x,y) = <* x*y *>
       proof
A1:      len LBZ(D,0,x,y) = 0+1 by Def4; then
         dom(LBZ(D,0,x,y)) = Seg 1 by FINSEQ_1:def 3; then
A2:      1 in dom(LBZ(D,0,x,y));
A3:      0 choose (1-'1) = 0 choose 0 by XREAL_1:232 .= 1 by NEWTON:19;
A4:      1-'1 = 0 by XREAL_1:232;
         LBZ(D,0,x,y).1 = 1*((D|^(0+1-'1)).x)*((D|^(1 -' 1)).y) by A3,A2,Def4
         .= ((D|^(0+0)).x)*((D|^0).y) by A4,BINOM:13
         .= ((D|^0).x)*((id R).y) by VECTSP11:18
         .= ((id R).x)*((id R).y) by VECTSP11:18
         .= x*y;
         hence thesis by A1,FINSEQ_1:40;
       end;
