reserve FS for non empty doubleLoopStr;
reserve F for Field;
reserve R for Abelian add-associative right_zeroed right_complementable non
  empty addLoopStr,
  x, y, z for Scalar of R;
reserve SF for Skew-Field,
  x, y, z for Scalar of SF;
reserve R, R1, R2 for Ring;
reserve R for Abelian add-associative right_zeroed right_complementable
  associative well-unital right_unital distributive non empty doubleLoopStr,
  F for non degenerated almost_left_invertible Ring,
  x for Scalar of F,
  V for add-associative right_zeroed right_complementable vector-distributive
  scalar-distributive scalar-associative scalar-unital
   non empty
  ModuleStr over F,
  v for Vector of V;
reserve V for add-associative right_zeroed right_complementable RightMod-like
  non empty RightModStr over R;
reserve x for Scalar of R;
reserve v,w for Vector of V;
reserve F for non degenerated almost_left_invertible Ring;
reserve x for Scalar of F;
reserve V for add-associative right_zeroed right_complementable RightMod-like
  non empty RightModStr over F;
reserve v for Vector of V;

theorem
  x<>0.F implies (v*x)*x"=v
proof
  assume
A1: x<>0.F;
  (v*x)*x" = v*(x*x") by Def8
    .= v* 1_F by A1,Th9
    .= v by Def8;
  hence thesis;
end;
