reserve X for non empty UNITSTR;
reserve a, b for Real;
reserve x, y for Point of X;
reserve X for RealUnitarySpace;
reserve x, y, z, u, v for Point of X;

theorem Th32:
  ||.x.|| - ||.y.|| <= ||.x - y.||
proof
  (x - y) + y = x - (y - y) by RLVECT_1:29
    .= x - 09(X) by RLVECT_1:15
    .= x by RLVECT_1:13;
  then ||.x.|| <= ||.x - y.|| + ||.y.|| by Th30;
  hence thesis by XREAL_1:20;
end;
