
theorem Th8:
  for L being add-associative right_zeroed right_complementable
        associative commutative left-distributive well-unital
        almost_left_invertible non empty doubleLoopStr
  for a,b,x being Element of L st b <> 0.L
  holds eval(<%a,b%>,-a/b) = 0.L
  proof
    let L be add-associative right_zeroed right_complementable
        associative commutative left-distributive well-unital
        almost_left_invertible non empty doubleLoopStr;
    let a,b,x be Element of L;
    assume b <> 0.L;
    then
A1: b*(/b) = 1.L by VECTSP_1:def 10;
    -a/b = (-a)/b by VECTSP_1:9;
    then b*(-a/b) = (-a)*1.L by A1,GROUP_1:def 3
    .= -a;
    hence eval(<%a,b%>,-a/b) = a+-a by POLYNOM5:44
    .= 0.L by RLVECT_1:5;
  end;
