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Spoiler for Not so sure but
Lets say there were x amount of water and wine. First we transfer y amount of water to wine. Now wine occupies x/(x+y) percent of mixture and water y/(x+y). When transfer back Y amount of mixture there goes xy/(x+y) amount of wine with it (and y^2/(x+y) water as well). And y-y^2/(x+y)=yx/(y+x) water stays in the second glass meaning amount of wine in the first glas is equal to water in the second bottle.
Another approach: if we put infinitesimal amount of water to wine and bring back equal amount of mixture obviously first glass would have infinitesimal amount of wine and second glass would have infintesimal amount of water. As opposite if we transfer all the water to second bottle and return half of the mixture of course there would be as much wine in the first glass as water in the second glass. And as we increase amount of transition; amount of water and wine in not original glasses will increase as well meaning this is continuous function (above mentioned regulatority will be present no matter how much liquid we transfer).
Another thing is there can never be more water in second bottle than wine and vice versa. So after every pair of transfer there is equal or more than 50% wine in the second glass and 50% or less wine in first glass. And if and if only we put all the liquid to second glass and transfer back the half we can have same amount of wine in both glass.
Another approach: if we put infinitesimal amount of water to wine and bring back equal amount of mixture obviously first glass would have infinitesimal amount of wine and second glass would have infintesimal amount of water. As opposite if we transfer all the water to second bottle and return half of the mixture of course there would be as much wine in the first glass as water in the second glass. And as we increase amount of transition; amount of water and wine in not original glasses will increase as well meaning this is continuous function (above mentioned regulatority will be present no matter how much liquid we transfer).
Another thing is there can never be more water in second bottle than wine and vice versa. So after every pair of transfer there is equal or more than 50% wine in the second glass and 50% or less wine in first glass. And if and if only we put all the liquid to second glass and transfer back the half we can have same amount of wine in both glass.
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