reserve F for Field,
  x for Element of F,
  V for VectSp of F,
  v for Element of V;

theorem Th20:
  for F being add-associative right_zeroed right_complementable
  commutative associative well-unital non degenerated almost_left_invertible
  distributive non empty doubleLoopStr, x being Element of F holds x <> 0.F
  implies (x")" = x
proof
  let F be add-associative right_zeroed right_complementable commutative
  associative well-unital non degenerated almost_left_invertible distributive
  non empty doubleLoopStr, x be Element of F;
  assume
A1: x <> 0.F;
  x <> 0.F implies x" <> 0.F
  proof
    assume
A2: x <> 0.F;
    assume not thesis;
    then 1.F = x*0.F by A2,Def10;
    hence contradiction;
  end;
  then x"*(x")" = 1.F by A1,Def10;
  then (x*x")*(x")" = x*1.F by GROUP_1:def 3;
  then 1.F*(x")" = x*1.F by A1,Def10;
  then (x")" = x*1.F;
  hence thesis;
end;
