Extension of positive operators

The main result states that if $E$ is a separable Fréchet lattice and $F$ is a (locally solid) topological vector lattice with the $\sigma$-interpolation property then each positive linear operator $T_0$ from a majorizing subspace $G\subset E$ into $F$ admits extension to a continuous positive linear operator $T$ from $E$ into $F$. This fact is proved by using only the axiom of countable choice.