We know that if the inner product of two functions $\phi_k$ and $\phi_n$ with respect to the weight function $w$ is zero, it is said that the functions are orthogonal: $$\int_0^L w(x) \phi_k(x) \phi_n(x)dx = \delta_{k,n} \tag1$$ where $\delta_{k,n}$ is the Kronecker delta.
However there is another definition "Bi-orthogonality" (e.g. of sine and cosine functions) https://mathworld.wolfram.com/GeneralizedFourierSeries.html which is used for two sets of mutually orthogonal functions $f_{1,k}$ and $f_{2,n}$: $$\int_0^L w(x)f_{1,k}(x)f_{1,n}(x)dx = \delta_{k,n} \tag2$$ $$\int_0^L w(x)f_{2,k}(x)f_{2,n}(x)dx = \delta_{k,n} \tag3$$ $$\int_0^L w(x)f_{1,k}(x)f_{2,n}(x)dx = 0 \tag4$$ I assume there is no difference. We can just combine $f_{1,k}$ and $f_{2,k}$ into $\phi_k$ as follows: $$ \phi_{2k-1} = f_{1,k} $$ $$ \phi_{2k} = f_{2,k} $$ and then (2), (3) and (4) are automatically combined in (1). Why don't we do this?