
theorem Th2:
  for L being add-associative right_zeroed right_complementable
  associative commutative well-unital almost_left_invertible distributive non
empty doubleLoopStr, x,y being Element of L st x <> 0.L & y <> 0.L holds (x *
  y)" = x" * y"
proof
  let L be add-associative right_zeroed right_complementable associative
  commutative well-unital almost_left_invertible distributive non empty
  doubleLoopStr;
  let x,y be Element of L;
  assume that
A1: x <> 0.L and
A2: y <> 0.L;
A3: (x" * y") * (x * y) = x" * y" * y * x by GROUP_1:def 3
    .= x" * (y" * y) * x by GROUP_1:def 3
    .= x" * 1.L * x by A2,VECTSP_1:def 10
    .= x" * x
    .= 1.L by A1,VECTSP_1:def 10;
  x * y <> 0.L by A1,A2,VECTSP_1:12;
  hence thesis by A3,VECTSP_1:def 10;
end;
