Copyright (c) 1999 Association of Mizar Users
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; definitions TARSKI; theorems EQREL_1, ZFMISC_1, T_1TOPSP, PARTIT1, BVFUNC_2, BVFUNC11, BVFUNC14, FUNCT_4, CQC_LANG, TARSKI, ENUMSET1, FUNCT_1, CANTOR_1, SETFAM_1, XBOOLE_0, XBOOLE_1; 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 Th1: G={A,B,C,D,E} & A<>B & A<>C & A<>D & A<>E implies CompF(A,G) = B '/\' C '/\' D '/\' E proof assume A1: G={A,B,C,D,E} & A<>B & A<>C & A<>D & A<>E; per cases; suppose A2: B = C; then G = {A,B,B,D} \/ {E} by A1,ENUMSET1:50 .= {B,B,A,D} \/ {E} by ENUMSET1:110 .= {B,B,A,D,E} by ENUMSET1:50 .= {B,A,D,E} by ENUMSET1:72 .={A,B,D,E} by ENUMSET1:108; hence CompF(A,G) = B '/\' D '/\' E by A1,BVFUNC14:7 .= B '/\' C '/\' D '/\' E by A2,PARTIT1:15; suppose A3: B = D; then G = {A,B,C,B} \/ {E} by A1,ENUMSET1:50 .= {B,B,A,C} \/ {E} by ENUMSET1:112 .= {B,B,A,C,E} by ENUMSET1:50 .= {B,A,C,E} by ENUMSET1:72 .={A,B,C,E} by ENUMSET1:108; hence CompF(A,G) = B '/\' C '/\' E by A1,BVFUNC14:7 .= B '/\' D '/\' C '/\' E by A3,PARTIT1:15 .= B '/\' C '/\' D '/\' E by PARTIT1:16; suppose A4: B = E; then G = {A} \/ {B,C,D,B} by A1,ENUMSET1:47 .= {A} \/ {B,B,C,D} by ENUMSET1:105 .= {A} \/ {B,C,D} by ENUMSET1:71 .={A,B,C,D} by ENUMSET1:44; hence CompF(A,G) = B '/\' C '/\' D by A1,BVFUNC14:7 .= B '/\' E '/\' C '/\' D by A4,PARTIT1:15 .= B '/\' E '/\' (C '/\' D) by PARTIT1:16 .= B '/\' (C '/\' D) '/\' E by PARTIT1:16 .= B '/\' C '/\' D '/\' E by PARTIT1:16; suppose A5: C = D; then G = {A,B,C,C} \/ {E} by A1,ENUMSET1:50 .= {C,C,A,B} \/ {E} by ENUMSET1:118 .= {C,A,B} \/ {E} by ENUMSET1:71 .= {C,A,B,E} by ENUMSET1:46 .={A,B,C,E} by ENUMSET1:110; hence CompF(A,G) = B '/\' C '/\' E by A1,BVFUNC14:7 .= B '/\' (C '/\' D) '/\' E by A5,PARTIT1:15 .= B '/\' C '/\' D '/\' E by PARTIT1:16; suppose A6: C = E; then G = {A} \/ {B,C,D,C} by A1,ENUMSET1:47 .= {A} \/ {C,C,B,D} by ENUMSET1:117 .= {A} \/ {C,B,D} by ENUMSET1:71 .= {A,C,B,D} by ENUMSET1:44 .={A,B,C,D} by ENUMSET1:104; hence CompF(A,G) = B '/\' C '/\' D by A1,BVFUNC14:7 .= B '/\' (C '/\' E) '/\' D by A6,PARTIT1:15 .= B '/\' (C '/\' E '/\' D) by PARTIT1:16 .= B '/\' (C '/\' D '/\' E) by PARTIT1:16 .= B '/\' (C '/\' D) '/\' E by PARTIT1:16 .= B '/\' C '/\' D '/\' E by PARTIT1:16; suppose A7: D = E; then G = {A} \/ {B,C,D,D} by A1,ENUMSET1:47 .= {A} \/ {D,D,B,C} by ENUMSET1:118 .= {A} \/ {D,B,C} by ENUMSET1:71 .= {A,D,B,C} by ENUMSET1:44 .={A,B,C,D} by ENUMSET1:105; hence CompF(A,G) = B '/\' C '/\' D by A1,BVFUNC14:7 .= B '/\' C '/\' (D '/\' E) by A7,PARTIT1:15 .= B '/\' (C '/\' (D '/\' E)) by PARTIT1:16 .= B '/\' (C '/\' D '/\' E) by PARTIT1:16 .= B '/\' (C '/\' D) '/\' E by PARTIT1:16 .= B '/\' C '/\' D '/\' E by PARTIT1:16; suppose A8: B<>C & B<>D & B<>E & C<>D & C<>E & D<>E; G \ {A}={A} \/ {B,C,D,E} \ {A} by A1,ENUMSET1:47; then A9:G \ {A} = ({A} \ {A}) \/ ({B,C,D,E} \ {A}) by XBOOLE_1:42; A10:not B in {A} by A1,TARSKI:def 1; A11:not C in {A} by A1,TARSKI:def 1; A12:not D in {A} by A1,TARSKI:def 1; A13:not E in {A} by A1,TARSKI:def 1; {B,C,D,E} \ {A} = ({B} \/ {C,D,E}) \ {A} by ENUMSET1:44 .= ({B} \ {A}) \/ ({C,D,E} \ {A}) by XBOOLE_1:42 .= {B} \/ ({C,D,E} \ {A}) by A10,ZFMISC_1:67 .= {B} \/ (({C} \/ {D,E}) \ {A}) by ENUMSET1:42 .= {B} \/ (({C} \ {A}) \/ ({D,E} \ {A})) by XBOOLE_1:42 .= {B} \/ (({C} \ {A}) \/ {D,E}) by A12,A13,ZFMISC_1:72 .= {B} \/ ({C} \/ {D,E}) by A11,ZFMISC_1:67 .= {B} \/ {C,D,E} by ENUMSET1:42; then A14:G \ {A} = ({A} \ {A}) \/ {B,C,D,E} by A9,ENUMSET1:44; A in {A} by TARSKI:def 1; then A15: {A} \ {A}={} by ZFMISC_1:68; A16:B '/\' C '/\' D '/\' E c= '/\' (G \ {A}) proof let x be set; assume A17:x in B '/\' C '/\' D '/\' E; then x in INTERSECTION(B '/\' C '/\' D,E) \ {{}} by PARTIT1:def 4; then x in INTERSECTION(B '/\' C '/\' D,E) & not x in {{}} by XBOOLE_0:def 4 ; then consider bcd,e being set such that A18: bcd in B '/\' C '/\' D & e in E & x = bcd /\ e by SETFAM_1:def 5; bcd in INTERSECTION(B '/\' C,D) \ {{}} by A18,PARTIT1:def 4; then bcd in INTERSECTION(B '/\' C,D) & not bcd in {{}} by XBOOLE_0:def 4; then consider bc,d being set such that A19: bc in B '/\' C & d in D & bcd = bc /\ d by SETFAM_1:def 5; bc in INTERSECTION(B,C) \ {{}} by A19,PARTIT1:def 4; then bc in INTERSECTION(B,C) & not bc in {{}} by XBOOLE_0:def 4; then consider b,c being set such that A20: b in B & c in C & bc = b /\ c by SETFAM_1:def 5; set h = (B .--> b) +* (C .--> c) +* (D .--> d) +* (E .--> e); dom ((B .--> b) +* (C .--> c)) = dom (B .--> b) \/ dom (C .--> c) by FUNCT_4:def 1; then dom ((B .--> b) +* (C .--> c) +* (D .--> d)) = dom (B .--> b) \/ dom (C .--> c) \/ dom (D .--> d) by FUNCT_4:def 1; then A21:dom ((B .--> b) +* (C .--> c) +* (D .--> d) +* (E .--> e)) = dom (B .--> b) \/ dom (C .--> c) \/ dom (D .--> d) \/ dom (E .--> e) by FUNCT_4:def 1; A22: dom (B .--> b) = {B} by CQC_LANG:5; A23: dom (C .--> c) = {C} by CQC_LANG:5; A24: dom (D .--> d) = {D} by CQC_LANG:5; A25: dom (E .--> e) = {E} by CQC_LANG:5; A26: dom ((B .--> b) +* (C .--> c) +* (D .--> d) +* (E .--> e)) = {B} \/ {C} \/ {D} \/ {E} by A21,A22,A23,A24,CQC_LANG:5 .= {B,C} \/ {D} \/ {E} by ENUMSET1:41 .= {B,C,D} \/ {E} by ENUMSET1:43 .= {B,C,D,E} by ENUMSET1:46; A27: h.D = d proof not D in dom (E .--> e) by A8,A25,TARSKI:def 1; then A28:h.D=((B .--> b) +* (C .--> c) +* (D .--> d)).D by FUNCT_4:12; D in dom (D .--> d) by A24,TARSKI:def 1; then h.D = (D .--> d).D by A28,FUNCT_4:14; hence thesis by CQC_LANG:6; end; A29: h.B = b proof A30:not B in dom (D .--> d) by A8,A24,TARSKI:def 1; not B in dom (E .--> e) by A8,A25,TARSKI:def 1; then h.B=((B .--> b) +* (C .--> c) +* (D .--> d)).B by FUNCT_4:12; then A31:h.B=((B .--> b) +* (C .--> c)).B by A30,FUNCT_4:12; not B in dom (C .--> c) by A8,A23,TARSKI:def 1; then h.B=(B .--> b).B by A31,FUNCT_4:12; hence thesis by CQC_LANG:6; end; A32: h.C = c proof A33:not C in dom (D .--> d) by A8,A24,TARSKI:def 1; not C in dom (E .--> e) by A8,A25,TARSKI:def 1; then h.C=((B .--> b) +* (C .--> c) +* (D .--> d)).C by FUNCT_4:12; then A34:h.C=((B .--> b) +* (C .--> c)).C by A33,FUNCT_4:12; C in dom (C .--> c) by A23,TARSKI:def 1; then h.C=(C .--> c).C by A34,FUNCT_4:14; hence thesis by CQC_LANG:6; end; A35: h.E = e proof E in dom (E .--> e) by A25,TARSKI:def 1; then h.E = (E .--> e).E by FUNCT_4:14; hence thesis by CQC_LANG:6; end; A36: for p being set st p in (G \ {A}) holds h.p in p proof let p be set; assume A37: p in (G \ {A}); now per cases by A14,A15,A37,ENUMSET1:18; case p=D; hence thesis by A19,A27; case p=B; hence thesis by A20,A29; case p=C; hence thesis by A20,A32; case p=E; hence thesis by A18,A35; end; hence thesis; end; A38: D in dom h by A26,ENUMSET1:19; A39: B in dom h by A26,ENUMSET1:19; A40: C in dom h by A26,ENUMSET1:19; A41: E in dom h by A26,ENUMSET1:19; A42:rng h c= {h.D,h.B,h.C,h.E} proof let t be set; assume t in rng h; then consider x1 being set such that A43: x1 in dom h & t = h.x1 by FUNCT_1:def 5; now per cases by A26,A43,ENUMSET1:18; case x1=D; hence thesis by A43,ENUMSET1:19; case x1=B; hence thesis by A43,ENUMSET1:19; case x1=C; hence thesis by A43,ENUMSET1:19; case x1=E; hence thesis by A43,ENUMSET1:19; end; hence thesis; end; A44: {h.D,h.B,h.C,h.E} c= rng h proof let t be set; assume A45: t in {h.D,h.B,h.C,h.E}; now per cases by A45,ENUMSET1:18; case t=h.D; hence thesis by A38,FUNCT_1:def 5; case t=h.B; hence thesis by A39,FUNCT_1:def 5; case t=h.C; hence thesis by A40,FUNCT_1:def 5; case t=h.E; hence thesis by A41,FUNCT_1:def 5; end; hence thesis; end; then A46: {h.D,h.B,h.C,h.E} = rng h by A42,XBOOLE_0:def 10; rng h c= bool Y proof let t be set; assume A47: t in rng h; now per cases by A42,A47,ENUMSET1:18; case t=h.D; hence thesis by A19,A27; case t=h.B; hence thesis by A20,A29; case t=h.C; hence thesis by A20,A32; case t=h.E; hence thesis by A18,A35; end; hence thesis; end; then reconsider F=rng h as Subset-Family of Y by SETFAM_1:def 7; reconsider h as Function; A48: rng h <> {} by A38,FUNCT_1:12; A49:x c= Intersect F proof let u be set; assume A50:u in x; A51: h.D in {h.D,h.B,h.C,h.E} by ENUMSET1:19; for y be set holds y in F implies u in y proof let y be set; assume A52: y in F; now per cases by A42,A52,ENUMSET1:18; case A53: y=h.D; u in d /\ ((b /\ c) /\ e) by A18,A19,A20,A50,XBOOLE_1:16; hence thesis by A27,A53,XBOOLE_0:def 3; case A54: y=h.B; u in (c /\ (d /\ b)) /\ e by A18,A19,A20,A50,XBOOLE_1:16; then u in c /\ ((d /\ b) /\ e) by XBOOLE_1:16; then u in c /\ ((d /\ e) /\ b) by XBOOLE_1:16; then u in (c /\ (d /\ e)) /\ b by XBOOLE_1:16; hence thesis by A29,A54,XBOOLE_0:def 3; case A55: y=h.C; u in (c /\ (b /\ d)) /\ e by A18,A19,A20,A50,XBOOLE_1:16; then u in c /\ (b /\ d /\ e) by XBOOLE_1:16; hence thesis by A32,A55,XBOOLE_0:def 3; case y=h.E; hence thesis by A18,A35,A50,XBOOLE_0:def 3; end; hence thesis; end; then u in meet F by A44,A51,SETFAM_1:def 1; hence thesis by A44,A51,CANTOR_1:def 3; end; Intersect F c= x proof let t be set; assume t in Intersect F; then A56: t in meet (rng h) by A48,CANTOR_1:def 3; h.B in rng h & h.C in rng h & h.D in rng h & h.E in rng h by A46,ENUMSET1:19; then A57: t in h.B & t in h.C & t in h.D & t in h.E by A56,SETFAM_1:def 1; then t in b /\ c by A29,A32,XBOOLE_0:def 3; then t in (b /\ c) /\ d by A27,A57,XBOOLE_0:def 3; hence thesis by A18,A19,A20,A35,A57,XBOOLE_0:def 3; end; then A58:x = Intersect F by A49,XBOOLE_0:def 10; x<>{} by A17,EQREL_1:def 6; hence thesis by A14,A15,A26,A36,A58,BVFUNC_2:def 1; end; '/\' (G \ {A}) c= B '/\' C '/\' D '/\' E proof let x be set; assume x in '/\' (G \ {A}); then consider h being Function, F being Subset-Family of Y such that A59: dom h=(G \ {A}) & rng h = F & (for d being set st d in (G \ {A}) holds h.d in d) & x=Intersect F & x<>{} by BVFUNC_2:def 1; B in (G \ {A}) & C in (G \ {A}) & D in (G \ {A}) & E in (G \ {A}) by A14,A15,ENUMSET1:19; then A60:h.B in B & h.C in C & h.D in D & h.E in E by A59; B in dom h & C in dom h & D in dom h & E in dom h by A14,A15,A59,ENUMSET1 :19; then A61:h.B in rng h & h.C in rng h & h.D in rng h & h.E in rng h by FUNCT_1:def 5; A62:not (x in {{}}) by A59,TARSKI:def 1; A63:h.B /\ h.C /\ h.D /\ h.E c= x proof let m be set; assume A64: m in h.B /\ h.C /\ h.D /\ h.E; then A65: m in h.B /\ h.C /\ h.D & m in h.E by XBOOLE_0:def 3; then A66: m in h.B /\ h.C & m in h.D by XBOOLE_0:def 3; A67:rng h c= {h.B,h.C,h.D,h.E} proof let u be set; assume u in rng h; then consider x1 being set such that A68: x1 in dom h & u = h.x1 by FUNCT_1:def 5; now per cases by A14,A15,A59,A68,ENUMSET1:18; case x1=B; hence thesis by A68,ENUMSET1:19; case x1=C; hence thesis by A68,ENUMSET1:19; case x1=D; hence thesis by A68,ENUMSET1:19; case x1=E; hence thesis by A68,ENUMSET1:19; end; hence thesis; end; for y being set holds y in rng h implies m in y proof let y be set; assume A69: y in rng h; now per cases by A67,A69,ENUMSET1:18; case y=h.B; hence thesis by A66,XBOOLE_0:def 3; case y=h.C; hence thesis by A66,XBOOLE_0:def 3; case y=h.D; hence thesis by A65,XBOOLE_0:def 3; case y=h.E; hence thesis by A64,XBOOLE_0:def 3; end; hence thesis; end; then m in meet (rng h) by A61,SETFAM_1:def 1; hence thesis by A59,A61,CANTOR_1:def 3; end; A70: x c= h.B /\ h.C /\ h.D /\ h.E proof let m be set; assume m in x; then m in meet (rng h) by A59,A61,CANTOR_1:def 3; then A71:m in h.B & m in h.C & m in h.D & m in h.E by A61,SETFAM_1:def 1; then m in h.B /\ h.C by XBOOLE_0:def 3; then m in h.B /\ h.C /\ h.D by A71,XBOOLE_0:def 3; hence thesis by A71,XBOOLE_0:def 3; end; then A72:((h.B /\ h.C) /\ h.D) /\ h.E = x by A63,XBOOLE_0:def 10; set mbc=h.B /\ h.C; set mbcd=(h.B /\ h.C) /\ h.D; mbcd<>{} by A59,A63,A70,XBOOLE_0:def 10; then A73:not (mbcd in {{}}) by TARSKI:def 1; mbc<>{} by A59,A63,A70,XBOOLE_0:def 10; then A74:not (mbc in {{}}) by TARSKI:def 1; mbc in INTERSECTION(B,C) by A60,SETFAM_1:def 5; then mbc in INTERSECTION(B,C) \ {{}} by A74,XBOOLE_0:def 4; then mbc in B '/\' C by PARTIT1:def 4; then mbcd in INTERSECTION(B '/\' C,D) by A60,SETFAM_1:def 5; then mbcd in INTERSECTION(B '/\' C,D) \ {{}} by A73,XBOOLE_0:def 4; then mbcd in B '/\' C '/\' D by PARTIT1:def 4; then x in INTERSECTION(B '/\' C '/\' D,E) by A60,A72,SETFAM_1:def 5; then x in INTERSECTION(B '/\' C '/\' D,E) \ {{}} by A62,XBOOLE_0:def 4; hence thesis by PARTIT1:def 4; end; then '/\' (G \ {A}) = B '/\' C '/\' D '/\' E by A16,XBOOLE_0:def 10; hence thesis by BVFUNC_2:def 7; end; theorem Th2: 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 proof assume A1: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; {A,B,C,D,E}={A,B} \/ {C,D,E} by ENUMSET1:48; then G={B,A,C,D,E} by A1,ENUMSET1:48; hence thesis by A1,Th1; end; theorem Th3: 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 proof assume A1: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; {A,B,C,D,E}={A,B,C} \/ {D,E} by ENUMSET1:49; then {A,B,C,D,E}={A} \/ {B,C} \/ {D,E} by ENUMSET1:42; then {A,B,C,D,E}={A,C,B} \/ {D,E} by ENUMSET1:42; then {A,B,C,D,E}={A,C} \/ {B} \/ {D,E} by ENUMSET1:43; then {A,B,C,D,E}={C,A,B} \/ {D,E} by ENUMSET1:43; then G={C,A,B,D,E} by A1,ENUMSET1:49; hence thesis by A1,Th1; end; theorem Th4: 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 proof assume A1: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; {A,B,C,D,E}={A,B} \/ {C,D,E} by ENUMSET1:48; then {A,B,C,D,E}={A,B} \/ ({C,D} \/ {E}) by ENUMSET1:43; then {A,B,C,D,E}={A,B} \/ {D,C,E} by ENUMSET1:43; then G={A,B,D,C,E} by A1,ENUMSET1:48; hence thesis by A1,Th3; end; theorem 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 proof assume A1: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; {A,B,C,D,E}={A,B,C} \/ {D,E} by ENUMSET1:49; then G={A,B,C,E,D} by A1,ENUMSET1:49; hence thesis by A1,Th4; end; theorem Th6: 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' proof let A,B,C,D,E be set; let h be Function; let A',B',C',D',E' be set; assume A1: 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'); A2: dom (C .--> C') = {C} by CQC_LANG:5; A3: dom (D .--> D') = {D} by CQC_LANG:5; A4: dom (E .--> E') = {E} by CQC_LANG:5; A5: dom (A .--> A') = {A} by CQC_LANG:5; A6: h.A = A' proof A in dom (A .--> A') by A5,TARSKI:def 1; then h.A = (A .--> A').A by A1,FUNCT_4:14; hence thesis by CQC_LANG:6; end; not E in dom (A .--> A') by A1,A5,TARSKI:def 1; then A7:h.E=((B .--> B') +* (C .--> C') +* (D .--> D') +* (E .--> E')).E by A1,FUNCT_4:12; A8: h.B = B' proof not B in dom (A .--> A') by A1,A5,TARSKI:def 1; then A9:h.B=((B .--> B') +* (C .--> C') +* (D .--> D') +* (E .--> E')).B by A1,FUNCT_4:12; not B in dom (E .--> E') by A1,A4,TARSKI:def 1; then A10:h.B=((B .--> B') +* (C .--> C') +* (D .--> D')).B by A9,FUNCT_4:12; not B in dom (D .--> D') by A1,A3,TARSKI:def 1; then A11: ((B .--> B') +* (C .--> C') +* (D .--> D')).B= ((B .--> B') +* (C .--> C')).B by FUNCT_4:12; not B in dom (C .--> C') by A1,A2,TARSKI:def 1; then h.B=(B .--> B').B by A10,A11,FUNCT_4:12; hence thesis by CQC_LANG:6; end; A12: h.C = C' proof not C in dom (A .--> A') by A1,A5,TARSKI:def 1; then A13:h.C=((B .--> B') +* (C .--> C') +* (D .--> D') +* (E .--> E')).C by A1,FUNCT_4:12; not C in dom (E .--> E') by A1,A4,TARSKI:def 1; then A14:h.C=((B .--> B') +* (C .--> C') +* (D .--> D')).C by A13,FUNCT_4:12; not C in dom (D .--> D') by A1,A3,TARSKI:def 1; then A15: ((B .--> B') +* (C .--> C') +* (D .--> D')).C= ((B .--> B') +* (C .--> C')).C by FUNCT_4:12; C in dom (C .--> C') by A2,TARSKI:def 1; then h.C=(C .--> C').C by A14,A15,FUNCT_4:14; hence thesis by CQC_LANG:6; end; A16: h.D = D' proof not D in dom (A .--> A') by A1,A5,TARSKI:def 1; then A17:h.D=((B .--> B') +* (C .--> C') +* (D .--> D') +* (E .--> E')).D by A1,FUNCT_4:12; not D in dom (E .--> E') by A1,A4,TARSKI:def 1; then A18:h.D=((B .--> B') +* (C .--> C') +* (D .--> D')).D by A17,FUNCT_4:12; D in dom (D .--> D') by A3,TARSKI:def 1; then h.D=(D .--> D').D by A18,FUNCT_4:14; hence thesis by CQC_LANG:6; end; h.E = E' proof E in dom (E .--> E') by A4,TARSKI:def 1; then h.E=(E .--> E').E by A7,FUNCT_4:14; hence thesis by CQC_LANG:6; end; hence thesis by A6,A8,A12,A16; end; theorem Th7: 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} proof let A,B,C,D,E be set; let h be Function; let A',B',C',D',E' be set; assume A1: h = (B .--> B') +* (C .--> C') +* (D .--> D') +* (E .--> E') +* (A .--> A'); dom ((B .--> B') +* (C .--> C')) = dom (B .--> B') \/ dom (C .--> C') by FUNCT_4:def 1; then dom ((B .--> B') +* (C .--> C') +* (D .--> D')) = dom (B .--> B') \/ dom (C .--> C') \/ dom (D .--> D') by FUNCT_4:def 1; then dom ((B .--> B') +* (C .--> C') +* (D .--> D') +* (E .--> E')) = dom (B .--> B') \/ dom (C .--> C') \/ dom (D .--> D') \/ dom (E .--> E') by FUNCT_4:def 1; then A2: dom ((B .--> B') +* (C .--> C') +* (D .--> D') +* (E .--> E') +* (A .--> A')) = dom (B .--> B') \/ dom (C .--> C') \/ dom (D .--> D') \/ dom (E .--> E') \/ dom (A .--> A') by FUNCT_4:def 1; A3: dom (B .--> B') = {B} by CQC_LANG:5; A4: dom (C .--> C') = {C} by CQC_LANG:5; A5: dom (D .--> D') = {D} by CQC_LANG:5; dom (E .--> E') = {E} by CQC_LANG:5; then dom ((B .--> B') +* (C .--> C') +* (D .--> D') +* (E .--> E') +* (A .--> A')) = {A} \/ (({B} \/ {C}) \/ {D} \/ {E}) by A2,A3,A4,A5,CQC_LANG:5 .= {A} \/ ({B,C} \/ {D} \/ {E}) by ENUMSET1:41 .= {A} \/ ({B,C,D} \/ {E}) by ENUMSET1:43 .= {A} \/ {B,C,D,E} by ENUMSET1:46 .= {A,B,C,D,E} by ENUMSET1:47; hence thesis by A1; end; theorem Th8: 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} proof let A,B,C,D,E be set; let h be Function; let A',B',C',D',E' be set; assume h = (B .--> B') +* (C .--> C') +* (D .--> D') +* (E .--> E') +* (A .--> A'); then A1: dom h = {A,B,C,D,E} by Th7; then A2: A in dom h by ENUMSET1:24; A3: B in dom h by A1,ENUMSET1:24; A4: C in dom h by A1,ENUMSET1:24; A5: D in dom h by A1,ENUMSET1:24; A6: E in dom h by A1,ENUMSET1:24; A7:rng h c= {h.A,h.B,h.C,h.D,h.E} proof let t be set; assume t in rng h; then consider x1 being set such that A8: x1 in dom h & t = h.x1 by FUNCT_1:def 5; now per cases by A1,A8,ENUMSET1:23; case x1=A; hence thesis by A8,ENUMSET1:24; case x1=B; hence thesis by A8,ENUMSET1:24; case x1=C; hence thesis by A8,ENUMSET1:24; case x1=D; hence thesis by A8,ENUMSET1:24; case x1=E; hence thesis by A8,ENUMSET1:24; end; hence thesis; end; {h.A,h.B,h.C,h.D,h.E} c= rng h proof let t be set; assume A9: t in {h.A,h.B,h.C,h.D,h.E}; now per cases by A9,ENUMSET1:23; case t=h.A; hence thesis by A2,FUNCT_1:def 5; case t=h.B; hence thesis by A3,FUNCT_1:def 5; case t=h.C; hence thesis by A4,FUNCT_1:def 5; case t=h.D; hence thesis by A5,FUNCT_1:def 5; case t=h.E; hence thesis by A6,FUNCT_1:def 5; end; hence thesis; end; hence thesis by A7,XBOOLE_0:def 10; end; theorem 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) proof let G be Subset of PARTITIONS(Y); let A,B,C,D,E be a_partition of Y; let z,u be Element of Y; let h be Function; assume A1: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; set GG=EqClass(u,B '/\' C '/\' D '/\' E); set h = (B .--> EqClass(u,B)) +* (C .--> EqClass(u,C)) +* (D .--> EqClass(u,D)) +* (E .--> EqClass(u,E)) +* (A .--> EqClass(z,A)); A2: dom h = G by A1,Th7; A3: h.A = EqClass(z,A) by A1,Th6; A4: h.B = EqClass(u,B) by A1,Th6; A5: h.C = EqClass(u,C) by A1,Th6; A6: h.D = EqClass(u,D) by A1,Th6; A7: h.E = EqClass(u,E) by A1,Th6; A8: for d being set st d in G holds h.d in d proof let d be set; assume A9: d in G; now per cases by A1,A9,ENUMSET1:23; case A10:d=A; h.A=EqClass(z,A) by A1,Th6; hence thesis by A10; case A11:d=B; h.B=EqClass(u,B) by A1,Th6; hence thesis by A11; case A12:d=C; h.C=EqClass(u,C) by A1,Th6; hence thesis by A12; case A13:d=D; h.D=EqClass(u,D) by A1,Th6; hence thesis by A13; case A14:d=E; h.E=EqClass(u,E) by A1,Th6; hence thesis by A14; end; hence thesis; end; A in dom h by A1,A2,ENUMSET1:24; then A15: h.A in rng h by FUNCT_1:def 5; B in dom h by A1,A2,ENUMSET1:24; then A16: h.B in rng h by FUNCT_1:def 5; C in dom h by A1,A2,ENUMSET1:24; then A17: h.C in rng h by FUNCT_1:def 5; D in dom h by A1,A2,ENUMSET1:24; then A18: h.D in rng h by FUNCT_1:def 5; E in dom h by A1,A2,ENUMSET1:24; then A19: h.E in rng h by FUNCT_1:def 5; A20:rng h = {h.A,h.B,h.C,h.D,h.E} by Th8; rng h c= bool Y proof let t be set; assume A21: t in rng h; now per cases by A20,A21,ENUMSET1:23; case t=h.A; then t=EqClass(z,A) by A1,Th6; hence thesis; case t=h.B; hence thesis by A4; case t=h.C; hence thesis by A5; case t=h.D; hence thesis by A6; case t=h.E; hence thesis by A7; end; hence thesis; end; then reconsider FF=rng h as Subset-Family of Y by SETFAM_1:def 7; (Intersect FF)<>{} by A1,A2,A8,BVFUNC_2:def 5; then consider m being set such that A22: m in Intersect FF by XBOOLE_0:def 1; m in meet FF by A15,A22,CANTOR_1:def 3; then A23: m in h.A & m in h.B & m in h.C & m in h.D & m in h.E by A15,A16,A17,A18,A19,SETFAM_1:def 1; GG = EqClass(u,B '/\' C '/\' D) /\ EqClass(u,E) by BVFUNC14:1; then A24: GG = EqClass(u,B '/\' C) /\ EqClass(u,D) /\ EqClass(u,E) by BVFUNC14:1; m in EqClass(u,B) /\ EqClass(u,C) by A4,A5,A23,XBOOLE_0:def 3; then m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) by A6,A23,XBOOLE_0 :def 3; then m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) /\ EqClass(u,E) by A7,A23,XBOOLE_0:def 3; then m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) /\ EqClass(u,E) /\ EqClass(z,A) by A3,A23,XBOOLE_0:def 3; then EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) /\ EqClass(u,E) meets EqClass(z,A) by XBOOLE_0:4; hence thesis by A24,BVFUNC14:1; end; theorem 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)) proof let G be Subset of PARTITIONS(Y); let A,B,C,D,E be a_partition of Y; let z,u be Element of Y; assume A1: 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); then A2:CompF(B,G) = A '/\' C '/\' D '/\' E by Th2; set H=EqClass(z,CompF(B,G)); set I=EqClass(z,A); set GG=EqClass(u,(((B '/\' C) '/\' D) '/\' E)); set h = (B .--> EqClass(u,B)) +* (C .--> EqClass(u,C)) +* (D .--> EqClass(u,D)) +* (E .--> EqClass(u,E)) +* (A .--> EqClass(z,A)); A3: dom h = G by A1,Th7; A4: h.A = EqClass(z,A) by A1,Th6; A5: h.B = EqClass(u,B) by A1,Th6; A6: h.C = EqClass(u,C) by A1,Th6; A7: h.D = EqClass(u,D) by A1,Th6; A8: h.E = EqClass(u,E) by A1,Th6; A9: for d being set st d in G holds h.d in d proof let d be set; assume A10: d in G; now per cases by A1,A10,ENUMSET1:23; case A11:d=A; h.A=EqClass(z,A) by A1,Th6; hence thesis by A11; case A12:d=B; h.B=EqClass(u,B) by A1,Th6; hence thesis by A12; case A13:d=C; h.C=EqClass(u,C) by A1,Th6; hence thesis by A13; case A14:d=D; h.D=EqClass(u,D) by A1,Th6; hence thesis by A14; case A15:d=E; h.E=EqClass(u,E) by A1,Th6; hence thesis by A15; end; hence thesis; end; A in dom h by A1,A3,ENUMSET1:24; then A16: h.A in rng h by FUNCT_1:def 5; B in dom h by A1,A3,ENUMSET1:24; then A17: h.B in rng h by FUNCT_1:def 5; C in dom h by A1,A3,ENUMSET1:24; then A18: h.C in rng h by FUNCT_1:def 5; D in dom h by A1,A3,ENUMSET1:24; then A19: h.D in rng h by FUNCT_1:def 5; E in dom h by A1,A3,ENUMSET1:24; then A20: h.E in rng h by FUNCT_1:def 5; A21:rng h = {h.A,h.B,h.C,h.D,h.E} by Th8; rng h c= bool Y proof let t be set; assume A22: t in rng h; now per cases by A21,A22,ENUMSET1:23; case t=h.A; then t=EqClass(z,A) by A1,Th6; hence thesis; case t=h.B; hence thesis by A5; case t=h.C; hence thesis by A6; case t=h.D; hence thesis by A7; case t=h.E; hence thesis by A8; end; hence thesis; end; then reconsider FF=rng h as Subset-Family of Y by SETFAM_1:def 7; (Intersect FF)<>{} by A1,A3,A9,BVFUNC_2:def 5; then consider m being set such that A23: m in Intersect FF by XBOOLE_0:def 1; m in meet FF by A16,A23,CANTOR_1:def 3; then A24: m in h.A & m in h.B & m in h.C & m in h.D & m in h.E by A16,A17,A18,A19,A20,SETFAM_1:def 1; GG = EqClass(u,B '/\' C '/\' D) /\ EqClass(u,E) by BVFUNC14:1; then A25: GG = EqClass(u,B '/\' C) /\ EqClass(u,D) /\ EqClass(u,E) by BVFUNC14:1; m in EqClass(u,B) /\ EqClass(u,C) by A5,A6,A24,XBOOLE_0:def 3; then m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) by A7,A24,XBOOLE_0 :def 3; then m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) /\ EqClass(u,E) by A8,A24,XBOOLE_0:def 3; then m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) /\ EqClass(u,E) /\ EqClass(z,A) by A4,A24,XBOOLE_0:def 3; then GG /\ I <> {} by A25,BVFUNC14:1; then consider p being set such that A26: p in GG /\ I by XBOOLE_0:def 1; reconsider p as Element of Y by A26; set K=EqClass(p,C '/\' D '/\' E); A27:p in K & K in C '/\' D '/\' E by T_1TOPSP:def 1; A28: p in GG & p in I by A26,XBOOLE_0:def 3; then p in I /\ K by A27,XBOOLE_0:def 3; then I /\ K in INTERSECTION(A,C '/\' D '/\' E) & not (I /\ K in {{}}) by SETFAM_1:def 5,TARSKI:def 1; then A29: I /\ K in INTERSECTION(A,C '/\' D '/\' E) \ {{}} by XBOOLE_0:def 4; set L=EqClass(z,C '/\' D '/\' E); GG = EqClass(u,((B '/\' (C '/\' D)) '/\' E)) by PARTIT1:16; then GG = EqClass(u,B '/\' (C '/\' D '/\' E)) by PARTIT1:16; then A30: GG c= L by A1,BVFUNC11:3; A31: p in GG by A26,XBOOLE_0:def 3; p in EqClass(p,C '/\' D '/\' E) by T_1TOPSP:def 1; then K meets L by A30,A31,XBOOLE_0:3; then K=L by T_1TOPSP:9; then A32:z in K by T_1TOPSP:def 1; z in I by T_1TOPSP:def 1; then A33:z in I /\ K by A32,XBOOLE_0:def 3; A34:z in H by T_1TOPSP:def 1; A '/\' (C '/\' D '/\' E) = A '/\' (C '/\' D) '/\' E by PARTIT1:16; then A '/\' (C '/\' D '/\' E) = A '/\' C '/\' D '/\' E by PARTIT1:16; then I /\ K in CompF(B,G) by A2,A29,PARTIT1:def 4; then I /\ K = H or I /\ K misses H by EQREL_1:def 6; then p in H by A27,A28,A33,A34,XBOOLE_0:3,def 3; then p in GG /\ H by A31,XBOOLE_0:def 3; then GG meets H by XBOOLE_0:4; hence thesis by A1,Th1; end;