environ vocabulary PARTIT1, EQREL_1, BVFUNC_2, BOOLE, SETFAM_1, CAT_1, FUNCT_4, RELAT_1, FUNCT_1, CANTOR_1, T_1TOPSP; notation TARSKI, XBOOLE_0, ENUMSET1, ZFMISC_1, SUBSET_1, SETFAM_1, RELAT_1, FUNCT_1, CQC_LANG, FUNCT_4, EQREL_1, PARTIT1, CANTOR_1, BVFUNC_1, BVFUNC_2; constructors BVFUNC_2, SETWISEO, CANTOR_1, FUNCT_7, BVFUNC_1; clusters SUBSET_1, PARTIT1, CQC_LANG, XBOOLE_0; requirements SUBSET, BOOLE; begin :: Chap. 1 Preliminaries reserve Y for non empty set, G for Subset of PARTITIONS(Y), A, B, C, D, E for a_partition of Y; theorem :: BVFUNC22:1 G={A,B,C,D,E} & A<>B & A<>C & A<>D & A<>E implies CompF(A,G) = B '/\' C '/\' D '/\' E; theorem :: BVFUNC22:2 G is independent & G={A,B,C,D,E} & A<>B & A<>C & A<>D & A<>E & B<>C & B<>D & B<>E & C<>D & C<>E & D<>E implies CompF(B,G) = A '/\' C '/\' D '/\' E; theorem :: BVFUNC22:3 G is independent & G={A,B,C,D,E} & A<>B & A<>C & A<>D & A<>E & B<>C & B<>D & B<>E & C<>D & C<>E & D<>E implies CompF(C,G) = A '/\' B '/\' D '/\' E; theorem :: BVFUNC22:4 G is independent & G={A,B,C,D,E} & A<>B & A<>C & A<>D & A<>E & B<>C & B<>D & B<>E & C<>D & C<>E & D<>E implies CompF(D,G) = A '/\' B '/\' C '/\' E; theorem :: BVFUNC22:5 G is independent & G={A,B,C,D,E} & A<>B & A<>C & A<>D & A<>E & B<>C & B<>D & B<>E & C<>D & C<>E & D<>E implies CompF(E,G) = A '/\' B '/\' C '/\' D; theorem :: BVFUNC22:6 for A,B,C,D,E being set, h being Function, A',B',C',D',E' being set st A<>B & A<>C & A<>D & A<>E & B<>C & B<>D & B<>E & C<>D & C<>E & D<>E & h = (B .--> B') +* (C .--> C') +* (D .--> D') +* (E .--> E') +* (A .--> A') holds h.A = A' & h.B = B' & h.C = C' & h.D = D' & h.E = E'; theorem :: BVFUNC22:7 for A,B,C,D,E being set, h being Function, A',B',C',D',E' being set st h = (B .--> B') +* (C .--> C') +* (D .--> D') +* (E .--> E') +* (A .--> A') holds dom h = {A,B,C,D,E}; theorem :: BVFUNC22:8 for A,B,C,D,E being set, h being Function, A',B',C',D',E' being set st h = (B .--> B') +* (C .--> C') +* (D .--> D') +* (E .--> E') +* (A .--> A') holds rng h = {h.A,h.B,h.C,h.D,h.E}; theorem :: BVFUNC22:9 for G being Subset of PARTITIONS(Y), A,B,C,D,E being a_partition of Y, z,u being Element of Y, h being Function st G is independent & G={A,B,C,D,E} & A<>B & A<>C & A<>D & A<>E & B<>C & B<>D & B<>E & C<>D & C<>E & D<>E holds EqClass(u,B '/\' C '/\' D '/\' E) meets EqClass(z,A); theorem :: BVFUNC22:10 for G being Subset of PARTITIONS(Y), A,B,C,D,E being a_partition of Y, z,u being Element of Y st G is independent & G={A,B,C,D,E} & A<>B & A<>C & A<>D & A<>E & B<>C & B<>D & B<>E & C<>D & C<>E & D<>E & EqClass(z,C '/\' D '/\' E)=EqClass(u,C '/\' D '/\' E) holds EqClass(u,CompF(A,G)) meets EqClass(z,CompF(B,G));