environ vocabulary FUNCT_2, MARGREL1, PARTIT1, EQREL_1, ZF_LANG, BVFUNC_2, BVFUNC_1, T_1TOPSP; notation XBOOLE_0, SUBSET_1, MARGREL1, VALUAT_1, FRAENKEL, EQREL_1, BVFUNC_1, BVFUNC_2; constructors BVFUNC_2, BVFUNC_1; clusters MARGREL1, VALUAT_1, FRAENKEL; begin :: Chap. 1 Predicate Calculus reserve Y for non empty set, a,b for Element of Funcs(Y,BOOLEAN), G for Subset of PARTITIONS(Y), A, B for a_partition of Y; theorem :: BVFUNC13:1 G is independent implies All('not' All(a,A,G),B,G) '<' 'not' All(All(a,B,G),A,G); theorem :: BVFUNC13:2 All(All('not' a,A,G),B,G) '<' 'not' All(All(a,B,G),A,G); theorem :: BVFUNC13:3 All('not' Ex(a,A,G),B,G) '<' 'not' All(All(a,B,G),A,G); theorem :: BVFUNC13:4 G is independent implies All(Ex('not' a,A,G),B,G) '<' 'not' All(All(a,B,G),A,G); canceled; theorem :: BVFUNC13:6 G is independent implies Ex(All('not' a,A,G),B,G) '<' 'not' All(All(a,B,G),A,G); theorem :: BVFUNC13:7 G is independent implies Ex('not' Ex(a,A,G),B,G) '<' 'not' All(All(a,B,G),A,G); canceled; theorem :: BVFUNC13:9 G is independent implies 'not' All(Ex(a,A,G),B,G) '<' 'not' Ex(All(a,B,G),A,G); theorem :: BVFUNC13:10 G is independent implies 'not' Ex(Ex(a,A,G),B,G) '<' 'not' Ex(All(a,B,G),A,G); theorem :: BVFUNC13:11 'not' Ex(Ex(a,A,G),B,G) '<' 'not' All(Ex(a,B,G),A,G); canceled 2; theorem :: BVFUNC13:14 G is independent implies 'not' Ex(All(a,A,G),B,G) '<' 'not' All(All(a,B,G),A,G); theorem :: BVFUNC13:15 G is independent implies 'not' All(Ex(a,A,G),B,G) '<' 'not' All(All(a,B,G),A,G); theorem :: BVFUNC13:16 'not' Ex(Ex(a,A,G),B,G) '<' 'not' All(All(a,B,G),A,G); theorem :: BVFUNC13:17 G is independent implies 'not' Ex(All(a,A,G),B,G) '<' Ex('not' All(a,B,G),A,G); theorem :: BVFUNC13:18 G is independent implies 'not' All(Ex(a,A,G),B,G) '<' Ex('not' All(a,B,G),A,G); theorem :: BVFUNC13:19 'not' Ex(Ex(a,A,G),B,G) '<' Ex('not' All(a,B,G),A,G); theorem :: BVFUNC13:20 G is independent implies 'not' All(Ex(a,A,G),B,G) '<' All('not' All(a,B,G),A,G); theorem :: BVFUNC13:21 G is independent implies 'not' Ex(Ex(a,A,G),B,G) '<' All('not' All(a,B,G),A,G); theorem :: BVFUNC13:22 'not' Ex(Ex(a,A,G),B,G) '<' Ex('not' Ex(a,B,G),A,G); theorem :: BVFUNC13:23 G is independent implies 'not' Ex(Ex(a,A,G),B,G) = All('not' Ex(a,B,G),A,G); theorem :: BVFUNC13:24 G is independent implies 'not' All(Ex(a,A,G),B,G) '<' Ex(Ex('not' a,B,G),A,G); theorem :: BVFUNC13:25 'not' Ex(Ex(a,A,G),B,G) '<' Ex(Ex('not' a,B,G),A,G); theorem :: BVFUNC13:26 G is independent implies 'not' All(Ex(a,A,G),B,G) '<' All(Ex('not' a,B,G),A,G); theorem :: BVFUNC13:27 G is independent implies 'not' Ex(Ex(a,A,G),B,G) '<' All(Ex('not' a,B,G),A,G); theorem :: BVFUNC13:28 'not' Ex(Ex(a,A,G),B,G) '<' Ex(All('not' a,B,G),A,G); theorem :: BVFUNC13:29 G is independent implies 'not' Ex(Ex(a,A,G),B,G) = All(All('not' a,B,G),A,G); theorem :: BVFUNC13:30 G is independent implies Ex('not' Ex(a,A,G),B,G) '<' 'not' Ex(All(a,B,G),A,G); theorem :: BVFUNC13:31 All('not' Ex(a,A,G),B,G) '<' 'not' Ex(All(a,B,G),A,G); theorem :: BVFUNC13:32 All('not' Ex(a,A,G),B,G) '<' 'not' All(Ex(a,B,G),A,G); theorem :: BVFUNC13:33 G is independent implies All('not' Ex(a,A,G),B,G) = 'not' Ex(Ex(a,B,G),A,G); theorem :: BVFUNC13:34 G is independent implies Ex('not' All(a,A,G),B,G) = Ex('not' All(a,B,G),A,G); theorem :: BVFUNC13:35 G is independent implies All('not' All(a,A,G),B,G) '<' Ex('not' All(a,B,G),A,G); theorem :: BVFUNC13:36 G is independent implies Ex('not' Ex(a,A,G),B,G) '<' Ex('not' All(a,B,G),A,G); theorem :: BVFUNC13:37 All('not' Ex(a,A,G),B,G) '<' Ex('not' All(a,B,G),A,G); theorem :: BVFUNC13:38 G is independent implies Ex('not' Ex(a,A,G),B,G) '<' All('not' All(a,B,G),A,G); theorem :: BVFUNC13:39 G is independent implies All('not' Ex(a,A,G),B,G) '<' All('not' All(a,B,G),A,G); theorem :: BVFUNC13:40 All('not' Ex(a,A,G),B,G) '<' Ex('not' Ex(a,B,G),A,G); theorem :: BVFUNC13:41 G is independent implies All('not' Ex(a,A,G),B,G) = All('not' Ex(a,B,G),A,G); theorem :: BVFUNC13:42 G is independent implies Ex('not' Ex(a,A,G),B,G) '<' Ex(Ex('not' a,B,G),A,G); theorem :: BVFUNC13:43 All('not' Ex(a,A,G),B,G) '<' Ex(Ex('not' a,B,G),A,G); theorem :: BVFUNC13:44 G is independent implies Ex('not' Ex(a,A,G),B,G) '<' All(Ex('not' a,B,G),A,G); theorem :: BVFUNC13:45 G is independent implies All('not' Ex(a,A,G),B,G) '<' All(Ex('not' a,B,G),A,G); theorem :: BVFUNC13:46 All('not' Ex(a,A,G),B,G) '<' Ex(All('not' a,B,G),A,G); theorem :: BVFUNC13:47 G is independent implies All('not' Ex(a,A,G),B,G) = All(All('not' a,B,G),A,G); theorem :: BVFUNC13:48 G is independent implies Ex(All('not' a,A,G),B,G) '<' 'not' Ex(All(a,B,G),A,G); theorem :: BVFUNC13:49 G is independent implies All(All('not' a,A,G),B,G) '<' 'not' Ex(All(a,B,G),A,G); theorem :: BVFUNC13:50 All(All('not' a,A,G),B,G) '<' 'not' All(Ex(a,B,G),A,G); theorem :: BVFUNC13:51 G is independent implies All(All('not' a,A,G),B,G) '<' 'not' Ex(Ex(a,B,G),A,G); theorem :: BVFUNC13:52 G is independent implies Ex(Ex('not' a,A,G),B,G) '<' Ex('not' All(a,B,G),A,G); theorem :: BVFUNC13:53 G is independent implies All(Ex('not' a,A,G),B,G) '<' Ex('not' All(a,B,G),A,G); theorem :: BVFUNC13:54 G is independent implies Ex(All('not' a,A,G),B,G) '<' Ex('not' All(a,B,G),A,G); theorem :: BVFUNC13:55 All(All('not' a,A,G),B,G) '<' Ex('not' All(a,B,G),A,G); theorem :: BVFUNC13:56 G is independent implies Ex(All('not' a,A,G),B,G) '<' All('not' All(a,B,G),A,G); theorem :: BVFUNC13:57 G is independent implies All(All('not' a,A,G),B,G) '<' All('not' All(a,B,G),A,G); theorem :: BVFUNC13:58 All(All('not' a,A,G),B,G) '<' Ex('not' Ex(a,B,G),A,G); theorem :: BVFUNC13:59 G is independent implies All(All('not' a,A,G),B,G) = All('not' Ex(a,B,G),A,G); canceled; theorem :: BVFUNC13:61 All(Ex('not' a,A,G),B,G) '<' Ex(Ex('not' a,B,G),A,G); theorem :: BVFUNC13:62 G is independent implies Ex(All('not' a,A,G),B,G) '<' Ex(Ex('not' a,B,G),A,G);