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;

theorem Th32:
  v*(0.R) = 0.V & v*(-1_R) = -v & (0.V)*x = 0.V
proof
  v + v*(0.R) = v* (1_R) + v*(0.R) by Def8
    .= v*((1_R) + (0.R)) by Def8
    .= v*(1_R) by RLVECT_1:4
    .= v by Def8
    .= v + 0.V by RLVECT_1:4;
  hence
A1: v*(0.R) = 0.V by RLVECT_1:8;
  v*(-(1_R)) + v = v*( -(1_R)) + v* (1_R) by Def8
    .= v*((-(1_R)) + (1_R)) by Def8
    .= 0.V by A1,RLVECT_1:5;
  then v*(-(1_R)) + (v + -v) = 0.V + -v by RLVECT_1:def 3;
  then 0.V + -v = v*(-(1_R)) + 0.V by RLVECT_1:5
    .= v*(-(1_R)) by RLVECT_1:4;
  hence v*(-1_R) = -v by RLVECT_1:4;
  (0.V)*x = v*((0.R) *x) by A1,Def8
    .= 0.V by A1;
  hence thesis;
end;
