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

theorem
  for F being Field, x being Element of F holds x <> 0.F implies
  x" <> 0.F & -x" <> 0.F
proof
  let F be Field, x be Element of F;
  assume
A1: x <> 0.F;
  hereby
    assume x" = 0.F;
    then 1.F = x*0.F by A1,Def10;
    hence contradiction;
  end;
  assume -x" = 0.F;
  then 1.F*x" = (-1.F)*0.F by Th6;
  then x*x" = x*0.F;
  then 1.F = x*0.F by A1,Def10;
  hence contradiction;
end;
