Consider the following theorem extracted from "An Introduction to Functional Analysis" by Charles Swartz (1992):
Theorem 1: Let $X$ be a vector space. Let $\mathcal{U}$ be a family of subsets of $X$ satisfying:
(i) each $U\in\mathcal{U}$ is balanced and absorbing,
(ii) if $U_1,U_2\in\mathcal{U},\exists U_3\in\mathcal{U}$ such that $U_3\subset U_1\cap U_2$,
(iii) if $U_1\in\mathcal{U}, \exists U_2\in\mathcal{U}$ such that $U_2+U_2\subset U_1$.
Then there exists a unique topology $\tau$ on $X$ such that $(X,\tau)$ is a TVS and $\mathcal{U}$ is a neighborhood base at $0$ (not necessarily of open sets) for $\tau$.
I am sure that a few years ago I saw the following theorem in some place but I cannot neither prove it nor find it:
Theorem 2: Let $X$ be a vector space over $K$. Let $\mathcal{U}$ be a family of subsets of $X$ satisfying:
(i) each $U\in\mathcal{U}$ is balanced and absorbing,
(ii) if $U_1,U_2\in\mathcal{U},\exists U_3\in\mathcal{U}$ such that $U_3\subset U_1\cap U_2$,
($\widehat{iii}$) if $U\in\mathcal{U}$ and $\lambda\in K,\lambda\neq0$, then $\lambda U\in\mathcal{U}$.
Then there exists a unique topology $\tau$ on $X$ such that $(X,\tau)$ is a TVS and $\mathcal{U}$ is a neighborhood base at $0$ (not necessarily of open sets) for $\tau$.
Question: How to prove that the sets of axioms $\{(i),(ii),(iii)\}$ and $\{(i),(ii),(\widehat{iii})\}$ are equivalent? In other words, how to prove the theorem 2?
In the proof of Theorem 1, $(ii)$ is enough to show that the family of all the sets $V$ satisfying $\forall x\in V,\exists U\in\mathcal{U},x+U\subset V$ is a topology. The condition $(iii)$ implies the continuity of the addition and then $(i),(ii)$ and $(iii)$ are used to prove the continuity of the scalar multiplication.
I have tried to do something analogous with $(i),(ii)$ and $(\widehat{iii})$ but without success.
Update: (i) implies that the maps $\lambda\mapsto\lambda x_0$ and $(\lambda,x)\mapsto \lambda x$ are continuous at $0\in K$ and at $(0,0)\in K\times X$ respectively.
$(\widehat{iii})$ implies that the map $x\mapsto\lambda_0x$ is continuous at $0\in X$. Then from the equality $$\lambda x-\lambda_0 x_0=(\lambda-\lambda_0)x_0+\lambda_0(x-x_0)+(\lambda-\lambda_0)(x-x_0)$$ we deduce that the scalar multiplication is continuous.
Question: How to prove that the addition is continuous?