The Similarity constraint implies: jw 0 j 0 sim(w 0 ; w ) = jv 0 j 0 sim(v 0 ; v ) jwj 0 sim(w 0 ; w ) = jvj 0 sim(v 0 ; v ) Because sim(w;w 0 ) min(jwj; jw 0 j) is always true: jw 0 j 0 j wj = max(jwj; jw 0 j) 0 min(jwj; jw 0 j) max(jwj; jw 0 j) 0 sim(w;w 0 ).
: 8w 2 3(A; M); 9w 0 2 3(A; M)nfwg = jw 0 j 0 j wj max(jw 0 j 0 sim(w;w 0 ); jwj 0 sim(w;w 0 )) max(jv 0 j 0 sim(v;v 0 ); jvj 0 sim(v;v 0 )) By taking k the maximum over all v !