To weaken the claim that "Price of vegetables is very high in city area in comparison to that in rural area," we need to find a fact that contradicts or undermines this claim. Let's analyze the options provided:

A. There is least demand for vegetable in city area.
This fact supports the claim that the price of vegetables is high in the city area because low demand could contribute to higher prices due to limited supply and higher transportation costs.

B. Vegetables are subject to downward price control in city area by the government.
This fact weakens the claim by suggesting that the government has implemented measures to control and regulate vegetable prices in the city area, which could result in lower prices.

C. City dwellers purchase vegetable produced only in the city area.
This fact does not weaken the claim but rather supports it. If city dwellers exclusively purchase vegetables produced in the city area, it implies a limited supply, which can contribute to higher prices.

D. Vegetables are mostly a common item of every meal of a city dweller.
This fact does not weaken the claim but rather supports it. If vegetables are a common item in every meal of city dwellers, it suggests high demand, which can contribute to higher prices.

E. Abundantly produced rural vegetables are not allowed to sell in the city area.
This fact weakens the claim by indicating that rural vegetables, which are abundantly produced and potentially cheaper, are not allowed to be sold in the city area. This restriction could contribute to higher prices in the city area.

Therefore, the fact that weakens the claim is option E: Abundantly produced rural vegetables are not allowed to sell in the city area.