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

theorem
  x*v = 0.V iff x = 0.F or v = 0.V
proof
  x*v = 0.V implies x = 0.F or v = 0.V
  proof
    assume x*v = 0.V;
    then
A1: x"*x*v = x"*(0.V) by Def15
      .= 0.V by Th10;
    assume x<>(0.F);
    then 0.V = (1.F)*v by A1,Def10;
    hence thesis;
  end;
  hence thesis by Th10;
end;
