reserve X for non empty set;
reserve x for Element of X;
reserve d1,d2 for Element of X;
reserve A for BinOp of X;
reserve M for Function of [:X,X:],X;
reserve V for Ring;
reserve V1 for Subset of V;
reserve V for Algebra;
reserve V1 for Subset of V;
reserve MR for Function of [:REAL,X:],X;
reserve a for Real;

theorem Th4:
  for V being add-associative right_zeroed right_complementable
  vector-distributive scalar-distributive scalar-associative
  vector-associative non empty AlgebraStr, a be Real holds a*0.V = 0.V
proof
  let V be add-associative right_zeroed right_complementable
  vector-distributive scalar-distributive scalar-associative
  vector-associative non
  empty AlgebraStr;
  let a be Real;
  a * 0.V + a * 0.V = a * (0.V + 0.V) by RLVECT_1:def 5;
  then a * 0.V + a * 0.V = a * 0.V;
  then a * 0.V + a * 0.V = a * 0.V + 0.V;
  hence thesis by RLVECT_1:8;
end;
