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
  dist(x,z) <= dist(x,y) + dist(y,z)
proof
  dist(x,z) = ||.(x-z)+09(X).|| by RLVECT_1:4
    .= ||.(x-z)+(y-y).|| by RLVECT_1:15
    .= ||.x-(z-(y-y)).|| by RLVECT_1:29
    .= ||.x-(y+(z-y)).|| by RLVECT_1:29
    .= ||.(x-y)-(z-y).|| by RLVECT_1:27
    .= ||.(x-y)+(y-z).|| by RLVECT_1:33;
  hence thesis by Th30;
end;
