
theorem
  for V be translatible Abelian add-associative right_zeroed
right_complementable triangle non empty RLSMetrStruct for v,w be Element of V
  holds Norm (v + w) <= Norm v + Norm w
proof
  let V be translatible Abelian add-associative right_zeroed
  right_complementable triangle non empty RLSMetrStruct;
  let v,w be Element of V;
  Norm (v + w) <= dist(0.V,v) + dist(v,v + w) by METRIC_1:4;
  then Norm (v + w) <= Norm v + dist(v + -v,v + w + -v) by Def6;
  then Norm (v + w) <= Norm v + dist(0.V,v + w + -v) by RLVECT_1:5;
  then Norm (v + w) <= Norm v + dist(0.V,w + ((-v) + v)) by RLVECT_1:def 3;
  then Norm (v + w) <= Norm v + dist(0.V,w + (0.V)) by RLVECT_1:5;
  hence thesis by RLVECT_1:4;
end;
