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How a player will win in this game, if you cant give check?

Lets say white is in turn 10, he cant check the oponent king. 
So, to white be able mate/capture the king in turn 12, black would need to
make changes the board position, by some movement and/or capture on turn
11, and that would then allow white capture king on turn 12. 
But the problem is: A player cant put his king in check, and so in turn 11
black would not be able to make this movement that put his king in check.

"Keller, in his 10th issue of World Game Review mentions the following paradox: what if, say white checks black, such that blacks only move is to check white, but in that position, whites only move is to check black, and so on and so on."
How this would happen in checkless chess? The rules say that you can only mate the enemy king, not check him. In fact this is the main idea of the variant.

Rodrigo, I think the idea is that if after white's move, black's only response will put white in (non-mate) check, then that black move is illegal, so black is in fact in checkmate, and white's move was legal.

Then, what happens if black's only response is to put white in a similar position?  This is the "paradox" that is referenced.

I independently came up with a near identical variant. Beaten to it by 200 years, dammit! However, my version avoids the paradox so is worth mentioning...

Yavalath Chess: If you check an opponent without checkmating them, you lose.

There is no 'paradox' here; just an ambiguity of the original rules, which can only be solved by more detailed specification on what constitutes a check. The same issue already occurs in orthodox Chess. In particular the question whether pieces that are pinned on their own King have there checking power subverted, or whether a King might step next to the opponent's King when he is protected. This is then solved through introduction of the concept 'pseudo-legal move', and stipulating that it is not allowed to expose your King to pseudo-legal capture. When one would have specified instead that a move is illegal when it exposes the player's own King to legal capture, this would be a circular definition. But that still doesn't make it meaningless; it just means it must be applied recursively. And fortunately that recursion always terminates, as there are only two Kings to capture. So the question really boils down to: in a contiguous sequence of questionable moves, will the first violation or the last violation be decisive.

Note that in Tai Shogi the opposit holds for the Emperor as in orthodox Chess for the King: Emperor's are allowed to move into each other's range, provided they are protected. (It must be, as the Emperor's range is the entire board!)

As to 'Yavalath Chess': note that the distinction between 'cannot' and 'loses if' is only relevant for determining whether stalemate is a draw or a loss, and has no bearing on the issue of what 'check' means.