Voting by Committees under Constraints

(1.2) Z C A∩B.

(1.3) MB (I,B)∩R_f=I,

where I = (Ли ({ж} U £>)) ∖Z and В = (B U ({ж} U D)) ∖Z.

Proof of Claim 1: Since TZ_f has the property that its minimal cylindric
decomposition has a unique section there exits G ∈ TZ_f and Z ∈ {{ж} , D}
such that ZGG and G = (G U ({ж} U D)) ∖Z TZ_f. Dehne

^MB(H, Z) = {B ∈ 2^κ I E = (B U ({ж} U £>)) ∖ZforE ∈ MB (H, Z)GTZ_f}

Denote ^Z = xAZ = Doτ^z=DAZ = x. Because G ∈ MB (G. Z) ∩
7⅞, then G ∈ ^MB (G, Z). Since G ¢ MB (G, ~ Z) Γ∖TZ_f then ^MB (G, Z) £
MB (G,~ Z) ∩ TZ_f . Let B be the element in the range with minimal
L_i-distance to Z with the property that MB (B, Z) MB (B, ~ Z) GTZ_f .
This implies that

^MB(B,Z)∖B = MB(B,{~Z}) Γ∖TZ_f.           (7)

Let A ∈ MB (B, Z) ∖B be such that MB (A, B) = {A, B}. Condition (??)
implies that A ∈ TZ_f and MB (A, B) ∩ TZ_f = A. This proves the Claim.

Let A, B ∈ TZ_f and Z ∈ {{ж} ,D} be such that conditions (1.1), (1-2),
and (1.3) of Claim 1 hold. Then we can generate, by an additive preference
with top on A U {~ Z}, the orderings A >∙ ^l B >∙ ^l A, A >² A >~² B, and
ʃl >- ³ ʃl >- ³ B, by an additive preference with top on BU{~ z}, the orderings
B>' A>' A and BM AM A, and by an additive preference with top
on B, the ordering A >∙⁶ B >∙⁶ A. Therefore, we have a free-triple on the
elements of the range A, B, and A, implying that here exists i ∈ N such that
w™ = wy = {{_i}}.

Case 2: Assume that for every D ∈ TZ_f such that у ∈ D, there exists BAD
such that B ∈ MB (D, 0) ∩ TZ_f.

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