Proof . ( of Lemma 10 ) ( 1 ) Given a confidence - rated predictor with inputs hypothesis set V k , unlabelled data U k , and error bound ǫ k /64 , the outputs { ( ξ k , i , ζ k , i , γ k , i ) } n k i=1 must satisfy that for all h , h ′ ∈ V k , [ I(h(x k , i ) = h ′ ( x k , i ) ) ( ξ k , i + ζ k , i ) ] ≤ ǫ k 32