
theorem Th8:
  for F being add-associative right_zeroed right_complementable
  associative commutative well-unital almost_left_invertible distributive non
  empty doubleLoopStr, x,y being Element of F holds x*y=0.F iff x=0.F or y=0.F
proof
  let F be add-associative right_zeroed right_complementable associative
  commutative well-unital almost_left_invertible distributive non empty
  doubleLoopStr, x,y be Element of F;
  x*y=0.F implies x=0.F or y=0.F
  proof
    assume
A1: x*y = 0.F;
    assume
A2: x<>0.F;
    x"*(0.F) = x"*x*y by A1,GROUP_1:def 3
      .= (1.F)*y by A2,Def10
      .= y;
    hence thesis;
  end;
  hence thesis;
end;
