reserve V for non empty RLSStruct;
reserve x,y,y1 for set;
reserve v for VECTOR of V;
reserve a,b for Real;

theorem Th19:
  for V being add-associative right_zeroed right_complementable
  scalar-distributive scalar-associative scalar-unital vector-distributive
  non empty RLSStruct,
  v being Element of V holds
  v = - v implies v = 0.V
proof
  let V be add-associative right_zeroed right_complementable
  scalar-distributive scalar-associative scalar-unital vector-distributive
  non empty RLSStruct,
  v be Element of V;
  assume v = - v;
  then 0.V = v + v by Def10
    .= 1 * v + v by Def8
    .= 1 * v + 1 * v by Def8
    .= (1 + 1) * v by Def6
    .= 2 * v;
  hence thesis by Th11;
end;
