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;

theorem
  x<>0.F implies x"*(x*v)=v
proof
  assume
A1: x<>0.F;
  x"*(x*v) = (x"*x)*v by VECTSP_1:def 16
    .= 1_F *v by A1,Th9
    .= v by VECTSP_1:def 17;
  hence thesis;
end;
