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, y - z) = dist(x,y)
proof
  thus dist(x - z,y - z) = ||.((x - z) - y) + z.|| by RLVECT_1:29
    .= ||.(x - (y + z)) + z.|| by RLVECT_1:27
    .= ||.((x - y) - z) + z.|| by RLVECT_1:27
    .= ||.(x - y) - (z - z).|| by RLVECT_1:29
    .= ||.(x - y) - 09(X).|| by RLVECT_1:15
    .= dist(x,y) by RLVECT_1:13;
end;
