For 2 people, division of good s is equitable, why following statement is true? As I said,
For 2 people, division of goods is equitable, then 
One of the following is true
1) The division is envy free
2) If they switch their shares, then the division is envy free.
I started by assuming that It is envy-free and tried to prove second statement is wrong. 
If it is envy-free and equitable, then two parties get same percentage of goods according to 
their valuation and no party values the other's shares more. 
So, If they switch their shares, one would envy the other's. So, it's not envy free? 
But my proof does not rely on "equitable" though..  

Let the people be $A$ and $B$. Let $v_A$ be $A$’s valuation of his share, and let $v_B$ be $B$’s valuation of his share. Saying that the division is equitable is saying that $v_A=v_B$. We can assume that the valuations range between $0$ and $1$ and represent the share of the goods that each thinks that he got.
There are three possibilities to consider:


*

*$v_A=v_B<\frac12$;  

*$v_A=v_B>\frac12$; and  

*$v_A=v_B=\frac12$.


In each case ask yourself two questions:


*

*Is this division envy-free?  

*What if they swap shares? Is the new division envy-free?

