:: BVFUNC14 semantic presentation begin theorem Th1: :: BVFUNC14:1 for Y being non empty set for z being Element of Y for PA, PB being a_partition of Y holds EqClass (z,(PA '/\' PB)) = (EqClass (z,PA)) /\ (EqClass (z,PB)) proof let Y be non empty set ; ::_thesis: for z being Element of Y for PA, PB being a_partition of Y holds EqClass (z,(PA '/\' PB)) = (EqClass (z,PA)) /\ (EqClass (z,PB)) let z be Element of Y; ::_thesis: for PA, PB being a_partition of Y holds EqClass (z,(PA '/\' PB)) = (EqClass (z,PA)) /\ (EqClass (z,PB)) let PA, PB be a_partition of Y; ::_thesis: EqClass (z,(PA '/\' PB)) = (EqClass (z,PA)) /\ (EqClass (z,PB)) A1: (EqClass (z,PA)) /\ (EqClass (z,PB)) c= EqClass (z,(PA '/\' PB)) proof set Z = EqClass (z,(PA '/\' PB)); let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (EqClass (z,PA)) /\ (EqClass (z,PB)) or x in EqClass (z,(PA '/\' PB)) ) assume A2: x in (EqClass (z,PA)) /\ (EqClass (z,PB)) ; ::_thesis: x in EqClass (z,(PA '/\' PB)) then reconsider x = x as Element of Y ; A3: x in EqClass (x,PA) by EQREL_1:def_6; x in EqClass (z,PA) by A2, XBOOLE_0:def_4; then A4: EqClass (x,PA) meets EqClass (z,PA) by A3, XBOOLE_0:3; A5: x in EqClass (x,PB) by EQREL_1:def_6; PA '/\' PB = (INTERSECTION (PA,PB)) \ {{}} by PARTIT1:def_4; then EqClass (z,(PA '/\' PB)) in INTERSECTION (PA,PB) by XBOOLE_0:def_5; then consider X, Y being set such that A6: X in PA and A7: Y in PB and A8: EqClass (z,(PA '/\' PB)) = X /\ Y by SETFAM_1:def_5; A9: z in X /\ Y by A8, EQREL_1:def_6; then ( z in EqClass (z,PB) & z in Y ) by EQREL_1:def_6, XBOOLE_0:def_4; then Y meets EqClass (z,PB) by XBOOLE_0:3; then A10: Y = EqClass (z,PB) by A7, EQREL_1:def_4; x in EqClass (z,PB) by A2, XBOOLE_0:def_4; then A11: EqClass (x,PB) meets EqClass (z,PB) by A5, XBOOLE_0:3; ( z in EqClass (z,PA) & z in X ) by A9, EQREL_1:def_6, XBOOLE_0:def_4; then X meets EqClass (z,PA) by XBOOLE_0:3; then X = EqClass (z,PA) by A6, EQREL_1:def_4; then A12: X = EqClass (x,PA) by A4, EQREL_1:41; x in (EqClass (x,PA)) /\ (EqClass (x,PB)) by A3, A5, XBOOLE_0:def_4; hence x in EqClass (z,(PA '/\' PB)) by A11, A8, A10, A12, EQREL_1:41; ::_thesis: verum end; EqClass (z,(PA '/\' PB)) c= (EqClass (z,PA)) /\ (EqClass (z,PB)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in EqClass (z,(PA '/\' PB)) or x in (EqClass (z,PA)) /\ (EqClass (z,PB)) ) A13: ( EqClass (z,(PA '/\' PB)) c= EqClass (z,PA) & EqClass (z,(PA '/\' PB)) c= EqClass (z,PB) ) by BVFUNC11:3; assume x in EqClass (z,(PA '/\' PB)) ; ::_thesis: x in (EqClass (z,PA)) /\ (EqClass (z,PB)) hence x in (EqClass (z,PA)) /\ (EqClass (z,PB)) by A13, XBOOLE_0:def_4; ::_thesis: verum end; hence EqClass (z,(PA '/\' PB)) = (EqClass (z,PA)) /\ (EqClass (z,PB)) by A1, XBOOLE_0:def_10; ::_thesis: verum end; theorem :: BVFUNC14:2 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G = {A,B} & A <> B holds '/\' G = A '/\' B proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B being a_partition of Y st G = {A,B} & A <> B holds '/\' G = A '/\' B let G be Subset of (PARTITIONS Y); ::_thesis: for A, B being a_partition of Y st G = {A,B} & A <> B holds '/\' G = A '/\' B let A, B be a_partition of Y; ::_thesis: ( G = {A,B} & A <> B implies '/\' G = A '/\' B ) assume that A1: G = {A,B} and A2: A <> B ; ::_thesis: '/\' G = A '/\' B A3: A '/\' B c= '/\' G proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in A '/\' B or x in '/\' G ) assume A4: x in A '/\' B ; ::_thesis: x in '/\' G then A5: x <> {} by EQREL_1:def_4; x in (INTERSECTION (A,B)) \ {{}} by A4, PARTIT1:def_4; then consider a, b being set such that A6: a in A and A7: b in B and A8: x = a /\ b by SETFAM_1:def_5; set h0 = (A,B) --> (a,b); A9: rng ((A,B) --> (a,b)) = {a,b} by A2, FUNCT_4:64; rng ((A,B) --> (a,b)) c= bool Y proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng ((A,B) --> (a,b)) or y in bool Y ) assume A10: y in rng ((A,B) --> (a,b)) ; ::_thesis: y in bool Y now__::_thesis:_(_(_y_=_a_&_y_in_bool_Y_)_or_(_y_=_b_&_y_in_bool_Y_)_) percases ( y = a or y = b ) by A9, A10, TARSKI:def_2; case y = a ; ::_thesis: y in bool Y hence y in bool Y by A6; ::_thesis: verum end; case y = b ; ::_thesis: y in bool Y hence y in bool Y by A7; ::_thesis: verum end; end; end; hence y in bool Y ; ::_thesis: verum end; then reconsider F = rng ((A,B) --> (a,b)) as Subset-Family of Y ; A11: x c= Intersect F proof let u be set ; :: according to TARSKI:def_3 ::_thesis: ( not u in x or u in Intersect F ) assume A12: u in x ; ::_thesis: u in Intersect F for y being set st y in F holds u in y proof let y be set ; ::_thesis: ( y in F implies u in y ) assume A13: y in F ; ::_thesis: u in y now__::_thesis:_(_(_y_=_a_&_u_in_y_)_or_(_y_=_b_&_u_in_y_)_) percases ( y = a or y = b ) by A9, A13, TARSKI:def_2; case y = a ; ::_thesis: u in y hence u in y by A8, A12, XBOOLE_0:def_4; ::_thesis: verum end; case y = b ; ::_thesis: u in y hence u in y by A8, A12, XBOOLE_0:def_4; ::_thesis: verum end; end; end; hence u in y ; ::_thesis: verum end; then u in meet F by A9, SETFAM_1:def_1; hence u in Intersect F by A9, SETFAM_1:def_9; ::_thesis: verum end; A14: for d being set st d in G holds ((A,B) --> (a,b)) . d in d proof let d be set ; ::_thesis: ( d in G implies ((A,B) --> (a,b)) . d in d ) assume A15: d in G ; ::_thesis: ((A,B) --> (a,b)) . d in d now__::_thesis:_(_(_d_=_A_&_((A,B)_-->_(a,b))_._d_in_d_)_or_(_d_=_B_&_((A,B)_-->_(a,b))_._d_in_d_)_) percases ( d = A or d = B ) by A1, A15, TARSKI:def_2; case d = A ; ::_thesis: ((A,B) --> (a,b)) . d in d hence ((A,B) --> (a,b)) . d in d by A2, A6, FUNCT_4:63; ::_thesis: verum end; case d = B ; ::_thesis: ((A,B) --> (a,b)) . d in d hence ((A,B) --> (a,b)) . d in d by A7, FUNCT_4:63; ::_thesis: verum end; end; end; hence ((A,B) --> (a,b)) . d in d ; ::_thesis: verum end; A16: rng ((A,B) --> (a,b)) = {a,b} by A2, FUNCT_4:64; Intersect F c= x proof let u be set ; :: according to TARSKI:def_3 ::_thesis: ( not u in Intersect F or u in x ) assume A17: u in Intersect F ; ::_thesis: u in x A18: a in {a,b} by TARSKI:def_2; then a in F by A2, FUNCT_4:64; then A19: Intersect F = meet F by SETFAM_1:def_9; b in {a,b} by TARSKI:def_2; then A20: u in b by A16, A17, A19, SETFAM_1:def_1; u in a by A16, A17, A18, A19, SETFAM_1:def_1; hence u in x by A8, A20, XBOOLE_0:def_4; ::_thesis: verum end; then ( dom ((A,B) --> (a,b)) = {A,B} & x = Intersect F ) by A11, FUNCT_4:62, XBOOLE_0:def_10; hence x in '/\' G by A1, A14, A5, BVFUNC_2:def_1; ::_thesis: verum end; '/\' G c= A '/\' B proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in '/\' G or x in A '/\' B ) assume x in '/\' G ; ::_thesis: x in A '/\' B then consider h being Function, F being Subset-Family of Y such that A21: dom h = G and A22: rng h = F and A23: for d being set st d in G holds h . d in d and A24: x = Intersect F and A25: x <> {} by BVFUNC_2:def_1; A26: not x in {{}} by A25, TARSKI:def_1; A in dom h by A1, A21, TARSKI:def_2; then A27: h . A in rng h by FUNCT_1:def_3; A28: (h . A) /\ (h . B) c= x proof let m be set ; :: according to TARSKI:def_3 ::_thesis: ( not m in (h . A) /\ (h . B) or m in x ) assume A29: m in (h . A) /\ (h . B) ; ::_thesis: m in x A30: rng h c= {(h . A),(h . B)} proof let u be set ; :: according to TARSKI:def_3 ::_thesis: ( not u in rng h or u in {(h . A),(h . B)} ) assume u in rng h ; ::_thesis: u in {(h . A),(h . B)} then consider x1 being set such that A31: x1 in dom h and A32: u = h . x1 by FUNCT_1:def_3; now__::_thesis:_(_(_x1_=_A_&_u_in_{(h_._A),(h_._B)}_)_or_(_x1_=_B_&_u_in_{(h_._A),(h_._B)}_)_) percases ( x1 = A or x1 = B ) by A1, A21, A31, TARSKI:def_2; case x1 = A ; ::_thesis: u in {(h . A),(h . B)} hence u in {(h . A),(h . B)} by A32, TARSKI:def_2; ::_thesis: verum end; case x1 = B ; ::_thesis: u in {(h . A),(h . B)} hence u in {(h . A),(h . B)} by A32, TARSKI:def_2; ::_thesis: verum end; end; end; hence u in {(h . A),(h . B)} ; ::_thesis: verum end; for y being set st y in rng h holds m in y proof let y be set ; ::_thesis: ( y in rng h implies m in y ) assume A33: y in rng h ; ::_thesis: m in y now__::_thesis:_(_(_y_=_h_._A_&_m_in_y_)_or_(_y_=_h_._B_&_m_in_y_)_) percases ( y = h . A or y = h . B ) by A30, A33, TARSKI:def_2; case y = h . A ; ::_thesis: m in y hence m in y by A29, XBOOLE_0:def_4; ::_thesis: verum end; case y = h . B ; ::_thesis: m in y hence m in y by A29, XBOOLE_0:def_4; ::_thesis: verum end; end; end; hence m in y ; ::_thesis: verum end; then m in meet (rng h) by A27, SETFAM_1:def_1; hence m in x by A22, A24, A27, SETFAM_1:def_9; ::_thesis: verum end; B in G by A1, TARSKI:def_2; then A34: h . B in B by A23; A in G by A1, TARSKI:def_2; then A35: h . A in A by A23; B in dom h by A1, A21, TARSKI:def_2; then A36: h . B in rng h by FUNCT_1:def_3; x c= (h . A) /\ (h . B) proof let m be set ; :: according to TARSKI:def_3 ::_thesis: ( not m in x or m in (h . A) /\ (h . B) ) assume m in x ; ::_thesis: m in (h . A) /\ (h . B) then m in meet (rng h) by A22, A24, A27, SETFAM_1:def_9; then ( m in h . A & m in h . B ) by A27, A36, SETFAM_1:def_1; hence m in (h . A) /\ (h . B) by XBOOLE_0:def_4; ::_thesis: verum end; then (h . A) /\ (h . B) = x by A28, XBOOLE_0:def_10; then x in INTERSECTION (A,B) by A35, A34, SETFAM_1:def_5; then x in (INTERSECTION (A,B)) \ {{}} by A26, XBOOLE_0:def_5; hence x in A '/\' B by PARTIT1:def_4; ::_thesis: verum end; hence '/\' G = A '/\' B by A3, XBOOLE_0:def_10; ::_thesis: verum end; Lm1: for B, C, D, b, c, d being set holds dom (((B .--> b) +* (C .--> c)) +* (D .--> d)) = {B,C,D} proof let B, C, D, b, c, d be set ; ::_thesis: dom (((B .--> b) +* (C .--> c)) +* (D .--> d)) = {B,C,D} A1: ( dom (C .--> c) = {C} & dom (D .--> d) = {D} ) by FUNCOP_1:13; ( dom ((B .--> b) +* (C .--> c)) = (dom (B .--> b)) \/ (dom (C .--> c)) & dom (B .--> b) = {B} ) by FUNCOP_1:13, FUNCT_4:def_1; hence dom (((B .--> b) +* (C .--> c)) +* (D .--> d)) = ({B} \/ {C}) \/ {D} by A1, FUNCT_4:def_1 .= {B,C} \/ {D} by ENUMSET1:1 .= {B,C,D} by ENUMSET1:3 ; ::_thesis: verum end; Lm2: for f being Function for C, D, c, d being set st C <> D holds ((f +* (C .--> c)) +* (D .--> d)) . C = c proof let f be Function; ::_thesis: for C, D, c, d being set st C <> D holds ((f +* (C .--> c)) +* (D .--> d)) . C = c let C, D, c, d be set ; ::_thesis: ( C <> D implies ((f +* (C .--> c)) +* (D .--> d)) . C = c ) set h = (f +* (C .--> c)) +* (D .--> d); A1: dom (D .--> d) = {D} by FUNCOP_1:13; assume C <> D ; ::_thesis: ((f +* (C .--> c)) +* (D .--> d)) . C = c then not C in dom (D .--> d) by A1, TARSKI:def_1; then A2: ((f +* (C .--> c)) +* (D .--> d)) . C = (f +* (C .--> c)) . C by FUNCT_4:11; dom (C .--> c) = {C} by FUNCOP_1:13; then C in dom (C .--> c) by TARSKI:def_1; hence ((f +* (C .--> c)) +* (D .--> d)) . C = (C .--> c) . C by A2, FUNCT_4:13 .= c by FUNCOP_1:72 ; ::_thesis: verum end; Lm3: for B, C, D, b, c, d being set st B <> C & D <> B holds (((B .--> b) +* (C .--> c)) +* (D .--> d)) . B = b proof let B, C, D, b, c, d be set ; ::_thesis: ( B <> C & D <> B implies (((B .--> b) +* (C .--> c)) +* (D .--> d)) . B = b ) assume that A1: B <> C and A2: D <> B ; ::_thesis: (((B .--> b) +* (C .--> c)) +* (D .--> d)) . B = b set h = ((B .--> b) +* (C .--> c)) +* (D .--> d); dom (D .--> d) = {D} by FUNCOP_1:13; then not B in dom (D .--> d) by A2, TARSKI:def_1; then A3: (((B .--> b) +* (C .--> c)) +* (D .--> d)) . B = ((B .--> b) +* (C .--> c)) . B by FUNCT_4:11; dom (C .--> c) = {C} by FUNCOP_1:13; then not B in dom (C .--> c) by A1, TARSKI:def_1; hence (((B .--> b) +* (C .--> c)) +* (D .--> d)) . B = (B .--> b) . B by A3, FUNCT_4:11 .= b by FUNCOP_1:72 ; ::_thesis: verum end; Lm4: for B, C, D, b, c, d being set for h being Function st h = ((B .--> b) +* (C .--> c)) +* (D .--> d) holds rng h = {(h . B),(h . C),(h . D)} proof let B, C, D, b, c, d be set ; ::_thesis: for h being Function st h = ((B .--> b) +* (C .--> c)) +* (D .--> d) holds rng h = {(h . B),(h . C),(h . D)} let h be Function; ::_thesis: ( h = ((B .--> b) +* (C .--> c)) +* (D .--> d) implies rng h = {(h . B),(h . C),(h . D)} ) assume h = ((B .--> b) +* (C .--> c)) +* (D .--> d) ; ::_thesis: rng h = {(h . B),(h . C),(h . D)} then A1: dom h = {B,C,D} by Lm1; then A2: B in dom h by ENUMSET1:def_1; A3: rng h c= {(h . B),(h . C),(h . D)} proof let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in rng h or t in {(h . B),(h . C),(h . D)} ) assume t in rng h ; ::_thesis: t in {(h . B),(h . C),(h . D)} then consider x1 being set such that A4: x1 in dom h and A5: t = h . x1 by FUNCT_1:def_3; now__::_thesis:_(_(_x1_=_D_&_t_in_{(h_._B),(h_._C),(h_._D)}_)_or_(_x1_=_B_&_t_in_{(h_._B),(h_._C),(h_._D)}_)_or_(_x1_=_C_&_t_in_{(h_._B),(h_._C),(h_._D)}_)_) percases ( x1 = D or x1 = B or x1 = C ) by A1, A4, ENUMSET1:def_1; case x1 = D ; ::_thesis: t in {(h . B),(h . C),(h . D)} hence t in {(h . B),(h . C),(h . D)} by A5, ENUMSET1:def_1; ::_thesis: verum end; case x1 = B ; ::_thesis: t in {(h . B),(h . C),(h . D)} hence t in {(h . B),(h . C),(h . D)} by A5, ENUMSET1:def_1; ::_thesis: verum end; case x1 = C ; ::_thesis: t in {(h . B),(h . C),(h . D)} hence t in {(h . B),(h . C),(h . D)} by A5, ENUMSET1:def_1; ::_thesis: verum end; end; end; hence t in {(h . B),(h . C),(h . D)} ; ::_thesis: verum end; A6: C in dom h by A1, ENUMSET1:def_1; A7: D in dom h by A1, ENUMSET1:def_1; {(h . B),(h . C),(h . D)} c= rng h proof let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in {(h . B),(h . C),(h . D)} or t in rng h ) assume A8: t in {(h . B),(h . C),(h . D)} ; ::_thesis: t in rng h now__::_thesis:_(_(_t_=_h_._D_&_t_in_rng_h_)_or_(_t_=_h_._B_&_t_in_rng_h_)_or_(_t_=_h_._C_&_t_in_rng_h_)_) percases ( t = h . D or t = h . B or t = h . C ) by A8, ENUMSET1:def_1; case t = h . D ; ::_thesis: t in rng h hence t in rng h by A7, FUNCT_1:def_3; ::_thesis: verum end; case t = h . B ; ::_thesis: t in rng h hence t in rng h by A2, FUNCT_1:def_3; ::_thesis: verum end; case t = h . C ; ::_thesis: t in rng h hence t in rng h by A6, FUNCT_1:def_3; ::_thesis: verum end; end; end; hence t in rng h ; ::_thesis: verum end; hence rng h = {(h . B),(h . C),(h . D)} by A3, XBOOLE_0:def_10; ::_thesis: verum end; theorem :: BVFUNC14:3 for Y being non empty set for G being Subset of (PARTITIONS Y) for B, C, D being a_partition of Y st G = {B,C,D} & B <> C & C <> D & D <> B holds '/\' G = (B '/\' C) '/\' D proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for B, C, D being a_partition of Y st G = {B,C,D} & B <> C & C <> D & D <> B holds '/\' G = (B '/\' C) '/\' D let G be Subset of (PARTITIONS Y); ::_thesis: for B, C, D being a_partition of Y st G = {B,C,D} & B <> C & C <> D & D <> B holds '/\' G = (B '/\' C) '/\' D let B, C, D be a_partition of Y; ::_thesis: ( G = {B,C,D} & B <> C & C <> D & D <> B implies '/\' G = (B '/\' C) '/\' D ) assume that A1: G = {B,C,D} and A2: B <> C and A3: C <> D and A4: D <> B ; ::_thesis: '/\' G = (B '/\' C) '/\' D A5: (B '/\' C) '/\' D c= '/\' G proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (B '/\' C) '/\' D or x in '/\' G ) assume A6: x in (B '/\' C) '/\' D ; ::_thesis: x in '/\' G then A7: x <> {} by EQREL_1:def_4; x in (INTERSECTION ((B '/\' C),D)) \ {{}} by A6, PARTIT1:def_4; then consider a, d being set such that A8: a in B '/\' C and A9: d in D and A10: x = a /\ d by SETFAM_1:def_5; a in (INTERSECTION (B,C)) \ {{}} by A8, PARTIT1:def_4; then consider b, c being set such that A11: b in B and A12: c in C and A13: a = b /\ c by SETFAM_1:def_5; set h = ((B .--> b) +* (C .--> c)) +* (D .--> d); A14: rng (((B .--> b) +* (C .--> c)) +* (D .--> d)) = {((((B .--> b) +* (C .--> c)) +* (D .--> d)) . B),((((B .--> b) +* (C .--> c)) +* (D .--> d)) . C),((((B .--> b) +* (C .--> c)) +* (D .--> d)) . D)} by Lm4 .= {((((B .--> b) +* (C .--> c)) +* (D .--> d)) . D),((((B .--> b) +* (C .--> c)) +* (D .--> d)) . B),((((B .--> b) +* (C .--> c)) +* (D .--> d)) . C)} by ENUMSET1:59 ; A15: (((B .--> b) +* (C .--> c)) +* (D .--> d)) . D = d by FUNCT_7:94; rng (((B .--> b) +* (C .--> c)) +* (D .--> d)) c= bool Y proof let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in rng (((B .--> b) +* (C .--> c)) +* (D .--> d)) or t in bool Y ) assume A16: t in rng (((B .--> b) +* (C .--> c)) +* (D .--> d)) ; ::_thesis: t in bool Y now__::_thesis:_(_(_t_=_(((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_._D_&_t_in_bool_Y_)_or_(_t_=_(((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_._B_&_t_in_bool_Y_)_or_(_t_=_(((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_._C_&_t_in_bool_Y_)_) percases ( t = (((B .--> b) +* (C .--> c)) +* (D .--> d)) . D or t = (((B .--> b) +* (C .--> c)) +* (D .--> d)) . B or t = (((B .--> b) +* (C .--> c)) +* (D .--> d)) . C ) by A14, A16, ENUMSET1:def_1; case t = (((B .--> b) +* (C .--> c)) +* (D .--> d)) . D ; ::_thesis: t in bool Y hence t in bool Y by A9, A15; ::_thesis: verum end; case t = (((B .--> b) +* (C .--> c)) +* (D .--> d)) . B ; ::_thesis: t in bool Y then t = b by A2, A4, Lm3; hence t in bool Y by A11; ::_thesis: verum end; case t = (((B .--> b) +* (C .--> c)) +* (D .--> d)) . C ; ::_thesis: t in bool Y then t = c by A3, Lm2; hence t in bool Y by A12; ::_thesis: verum end; end; end; hence t in bool Y ; ::_thesis: verum end; then reconsider F = rng (((B .--> b) +* (C .--> c)) +* (D .--> d)) as Subset-Family of Y ; A17: (((B .--> b) +* (C .--> c)) +* (D .--> d)) . C = c by A3, Lm2; A18: for p being set st p in G holds (((B .--> b) +* (C .--> c)) +* (D .--> d)) . p in p proof let p be set ; ::_thesis: ( p in G implies (((B .--> b) +* (C .--> c)) +* (D .--> d)) . p in p ) assume A19: p in G ; ::_thesis: (((B .--> b) +* (C .--> c)) +* (D .--> d)) . p in p now__::_thesis:_(_(_p_=_D_&_(((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_._p_in_p_)_or_(_p_=_B_&_(((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_._p_in_p_)_or_(_p_=_C_&_(((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_._p_in_p_)_) percases ( p = D or p = B or p = C ) by A1, A19, ENUMSET1:def_1; case p = D ; ::_thesis: (((B .--> b) +* (C .--> c)) +* (D .--> d)) . p in p hence (((B .--> b) +* (C .--> c)) +* (D .--> d)) . p in p by A9, FUNCT_7:94; ::_thesis: verum end; case p = B ; ::_thesis: (((B .--> b) +* (C .--> c)) +* (D .--> d)) . p in p hence (((B .--> b) +* (C .--> c)) +* (D .--> d)) . p in p by A2, A4, A11, Lm3; ::_thesis: verum end; case p = C ; ::_thesis: (((B .--> b) +* (C .--> c)) +* (D .--> d)) . p in p hence (((B .--> b) +* (C .--> c)) +* (D .--> d)) . p in p by A3, A12, Lm2; ::_thesis: verum end; end; end; hence (((B .--> b) +* (C .--> c)) +* (D .--> d)) . p in p ; ::_thesis: verum end; A20: (((B .--> b) +* (C .--> c)) +* (D .--> d)) . B = b by A2, A4, Lm3; A21: x c= Intersect F proof let u be set ; :: according to TARSKI:def_3 ::_thesis: ( not u in x or u in Intersect F ) assume A22: u in x ; ::_thesis: u in Intersect F for y being set st y in F holds u in y proof let y be set ; ::_thesis: ( y in F implies u in y ) assume A23: y in F ; ::_thesis: u in y now__::_thesis:_(_(_y_=_(((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_._D_&_u_in_y_)_or_(_y_=_(((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_._B_&_u_in_y_)_or_(_y_=_(((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_._C_&_u_in_y_)_) percases ( y = (((B .--> b) +* (C .--> c)) +* (D .--> d)) . D or y = (((B .--> b) +* (C .--> c)) +* (D .--> d)) . B or y = (((B .--> b) +* (C .--> c)) +* (D .--> d)) . C ) by A14, A23, ENUMSET1:def_1; case y = (((B .--> b) +* (C .--> c)) +* (D .--> d)) . D ; ::_thesis: u in y hence u in y by A10, A15, A22, XBOOLE_0:def_4; ::_thesis: verum end; caseA24: y = (((B .--> b) +* (C .--> c)) +* (D .--> d)) . B ; ::_thesis: u in y u in b /\ (c /\ d) by A10, A13, A22, XBOOLE_1:16; hence u in y by A20, A24, XBOOLE_0:def_4; ::_thesis: verum end; caseA25: y = (((B .--> b) +* (C .--> c)) +* (D .--> d)) . C ; ::_thesis: u in y u in c /\ (b /\ d) by A10, A13, A22, XBOOLE_1:16; hence u in y by A17, A25, XBOOLE_0:def_4; ::_thesis: verum end; end; end; hence u in y ; ::_thesis: verum end; then u in meet F by A14, SETFAM_1:def_1; hence u in Intersect F by A14, SETFAM_1:def_9; ::_thesis: verum end; A26: dom (((B .--> b) +* (C .--> c)) +* (D .--> d)) = {B,C,D} by Lm1; then D in dom (((B .--> b) +* (C .--> c)) +* (D .--> d)) by ENUMSET1:def_1; then A27: rng (((B .--> b) +* (C .--> c)) +* (D .--> d)) <> {} by FUNCT_1:3; Intersect F c= x proof let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in Intersect F or t in x ) assume t in Intersect F ; ::_thesis: t in x then A28: t in meet (rng (((B .--> b) +* (C .--> c)) +* (D .--> d))) by A27, SETFAM_1:def_9; (((B .--> b) +* (C .--> c)) +* (D .--> d)) . C in {((((B .--> b) +* (C .--> c)) +* (D .--> d)) . D),((((B .--> b) +* (C .--> c)) +* (D .--> d)) . B),((((B .--> b) +* (C .--> c)) +* (D .--> d)) . C)} by ENUMSET1:def_1; then t in (((B .--> b) +* (C .--> c)) +* (D .--> d)) . C by A14, A28, SETFAM_1:def_1; then A29: t in c by A3, Lm2; (((B .--> b) +* (C .--> c)) +* (D .--> d)) . B in {((((B .--> b) +* (C .--> c)) +* (D .--> d)) . D),((((B .--> b) +* (C .--> c)) +* (D .--> d)) . B),((((B .--> b) +* (C .--> c)) +* (D .--> d)) . C)} by ENUMSET1:def_1; then t in (((B .--> b) +* (C .--> c)) +* (D .--> d)) . B by A14, A28, SETFAM_1:def_1; then t in b by A2, A4, Lm3; then A30: t in b /\ c by A29, XBOOLE_0:def_4; (((B .--> b) +* (C .--> c)) +* (D .--> d)) . D in {((((B .--> b) +* (C .--> c)) +* (D .--> d)) . D),((((B .--> b) +* (C .--> c)) +* (D .--> d)) . B),((((B .--> b) +* (C .--> c)) +* (D .--> d)) . C)} by ENUMSET1:def_1; then t in (((B .--> b) +* (C .--> c)) +* (D .--> d)) . D by A14, A28, SETFAM_1:def_1; hence t in x by A10, A13, A15, A30, XBOOLE_0:def_4; ::_thesis: verum end; then x = Intersect F by A21, XBOOLE_0:def_10; hence x in '/\' G by A1, A26, A18, A7, BVFUNC_2:def_1; ::_thesis: verum end; '/\' G c= (B '/\' C) '/\' D proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in '/\' G or x in (B '/\' C) '/\' D ) assume x in '/\' G ; ::_thesis: x in (B '/\' C) '/\' D then consider h being Function, F being Subset-Family of Y such that A31: dom h = G and A32: rng h = F and A33: for d being set st d in G holds h . d in d and A34: x = Intersect F and A35: x <> {} by BVFUNC_2:def_1; D in dom h by A1, A31, ENUMSET1:def_1; then A36: h . D in rng h by FUNCT_1:def_3; set m = (h . B) /\ (h . C); B in dom h by A1, A31, ENUMSET1:def_1; then A37: h . B in rng h by FUNCT_1:def_3; C in dom h by A1, A31, ENUMSET1:def_1; then A38: h . C in rng h by FUNCT_1:def_3; A39: x c= ((h . B) /\ (h . C)) /\ (h . D) proof let m be set ; :: according to TARSKI:def_3 ::_thesis: ( not m in x or m in ((h . B) /\ (h . C)) /\ (h . D) ) assume m in x ; ::_thesis: m in ((h . B) /\ (h . C)) /\ (h . D) then A40: m in meet (rng h) by A32, A34, A37, SETFAM_1:def_9; then ( m in h . B & m in h . C ) by A37, A38, SETFAM_1:def_1; then A41: m in (h . B) /\ (h . C) by XBOOLE_0:def_4; m in h . D by A36, A40, SETFAM_1:def_1; hence m in ((h . B) /\ (h . C)) /\ (h . D) by A41, XBOOLE_0:def_4; ::_thesis: verum end; then (h . B) /\ (h . C) <> {} by A35; then A42: not (h . B) /\ (h . C) in {{}} by TARSKI:def_1; D in G by A1, ENUMSET1:def_1; then A43: h . D in D by A33; A44: not x in {{}} by A35, TARSKI:def_1; C in G by A1, ENUMSET1:def_1; then A45: h . C in C by A33; B in G by A1, ENUMSET1:def_1; then h . B in B by A33; then (h . B) /\ (h . C) in INTERSECTION (B,C) by A45, SETFAM_1:def_5; then (h . B) /\ (h . C) in (INTERSECTION (B,C)) \ {{}} by A42, XBOOLE_0:def_5; then A46: (h . B) /\ (h . C) in B '/\' C by PARTIT1:def_4; ((h . B) /\ (h . C)) /\ (h . D) c= x proof let m be set ; :: according to TARSKI:def_3 ::_thesis: ( not m in ((h . B) /\ (h . C)) /\ (h . D) or m in x ) assume A47: m in ((h . B) /\ (h . C)) /\ (h . D) ; ::_thesis: m in x then A48: m in (h . B) /\ (h . C) by XBOOLE_0:def_4; A49: rng h c= {(h . B),(h . C),(h . D)} proof let u be set ; :: according to TARSKI:def_3 ::_thesis: ( not u in rng h or u in {(h . B),(h . C),(h . D)} ) assume u in rng h ; ::_thesis: u in {(h . B),(h . C),(h . D)} then consider x1 being set such that A50: x1 in dom h and A51: u = h . x1 by FUNCT_1:def_3; now__::_thesis:_(_(_x1_=_B_&_u_in_{(h_._B),(h_._C),(h_._D)}_)_or_(_x1_=_C_&_u_in_{(h_._B),(h_._C),(h_._D)}_)_or_(_x1_=_D_&_u_in_{(h_._B),(h_._C),(h_._D)}_)_) percases ( x1 = B or x1 = C or x1 = D ) by A1, A31, A50, ENUMSET1:def_1; case x1 = B ; ::_thesis: u in {(h . B),(h . C),(h . D)} hence u in {(h . B),(h . C),(h . D)} by A51, ENUMSET1:def_1; ::_thesis: verum end; case x1 = C ; ::_thesis: u in {(h . B),(h . C),(h . D)} hence u in {(h . B),(h . C),(h . D)} by A51, ENUMSET1:def_1; ::_thesis: verum end; case x1 = D ; ::_thesis: u in {(h . B),(h . C),(h . D)} hence u in {(h . B),(h . C),(h . D)} by A51, ENUMSET1:def_1; ::_thesis: verum end; end; end; hence u in {(h . B),(h . C),(h . D)} ; ::_thesis: verum end; for y being set st y in rng h holds m in y proof let y be set ; ::_thesis: ( y in rng h implies m in y ) assume A52: y in rng h ; ::_thesis: m in y now__::_thesis:_(_(_y_=_h_._B_&_m_in_y_)_or_(_y_=_h_._C_&_m_in_y_)_or_(_y_=_h_._D_&_m_in_y_)_) percases ( y = h . B or y = h . C or y = h . D ) by A49, A52, ENUMSET1:def_1; case y = h . B ; ::_thesis: m in y hence m in y by A48, XBOOLE_0:def_4; ::_thesis: verum end; case y = h . C ; ::_thesis: m in y hence m in y by A48, XBOOLE_0:def_4; ::_thesis: verum end; case y = h . D ; ::_thesis: m in y hence m in y by A47, XBOOLE_0:def_4; ::_thesis: verum end; end; end; hence m in y ; ::_thesis: verum end; then m in meet (rng h) by A37, SETFAM_1:def_1; hence m in x by A32, A34, A37, SETFAM_1:def_9; ::_thesis: verum end; then ((h . B) /\ (h . C)) /\ (h . D) = x by A39, XBOOLE_0:def_10; then x in INTERSECTION ((B '/\' C),D) by A43, A46, SETFAM_1:def_5; then x in (INTERSECTION ((B '/\' C),D)) \ {{}} by A44, XBOOLE_0:def_5; hence x in (B '/\' C) '/\' D by PARTIT1:def_4; ::_thesis: verum end; hence '/\' G = (B '/\' C) '/\' D by A5, XBOOLE_0:def_10; ::_thesis: verum end; theorem Th4: :: BVFUNC14:4 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C being a_partition of Y st G = {A,B,C} & A <> B & C <> A holds CompF (A,G) = B '/\' C proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C being a_partition of Y st G = {A,B,C} & A <> B & C <> A holds CompF (A,G) = B '/\' C let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C being a_partition of Y st G = {A,B,C} & A <> B & C <> A holds CompF (A,G) = B '/\' C let A, B, C be a_partition of Y; ::_thesis: ( G = {A,B,C} & A <> B & C <> A implies CompF (A,G) = B '/\' C ) assume that A1: G = {A,B,C} and A2: A <> B and A3: C <> A ; ::_thesis: CompF (A,G) = B '/\' C percases ( B = C or B <> C ) ; supposeA4: B = C ; ::_thesis: CompF (A,G) = B '/\' C G = {B,C,A} by A1, ENUMSET1:59 .= {B,A} by A4, ENUMSET1:30 ; hence CompF (A,G) = B by A2, BVFUNC11:7 .= B '/\' C by A4, PARTIT1:13 ; ::_thesis: verum end; supposeA5: B <> C ; ::_thesis: CompF (A,G) = B '/\' C A6: G \ {A} = ({A} \/ {B,C}) \ {A} by A1, ENUMSET1:2 .= ({A} \ {A}) \/ ({B,C} \ {A}) by XBOOLE_1:42 ; ( not B in {A} & not C in {A} ) by A2, A3, TARSKI:def_1; then A7: G \ {A} = ({A} \ {A}) \/ {B,C} by A6, ZFMISC_1:63 .= {} \/ {B,C} by XBOOLE_1:37 .= {B,C} ; A8: '/\' (G \ {A}) c= B '/\' C proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in '/\' (G \ {A}) or x in B '/\' C ) assume x in '/\' (G \ {A}) ; ::_thesis: x in B '/\' C then consider h being Function, F being Subset-Family of Y such that A9: dom h = G \ {A} and A10: rng h = F and A11: for d being set st d in G \ {A} holds h . d in d and A12: x = Intersect F and A13: x <> {} by BVFUNC_2:def_1; A14: not x in {{}} by A13, TARSKI:def_1; B in dom h by A7, A9, TARSKI:def_2; then A15: h . B in rng h by FUNCT_1:def_3; A16: (h . B) /\ (h . C) c= x proof let m be set ; :: according to TARSKI:def_3 ::_thesis: ( not m in (h . B) /\ (h . C) or m in x ) assume A17: m in (h . B) /\ (h . C) ; ::_thesis: m in x A18: rng h c= {(h . B),(h . C)} proof let u be set ; :: according to TARSKI:def_3 ::_thesis: ( not u in rng h or u in {(h . B),(h . C)} ) assume u in rng h ; ::_thesis: u in {(h . B),(h . C)} then consider x1 being set such that A19: x1 in dom h and A20: u = h . x1 by FUNCT_1:def_3; now__::_thesis:_(_(_x1_=_B_&_u_in_{(h_._B),(h_._C)}_)_or_(_x1_=_C_&_u_in_{(h_._B),(h_._C)}_)_) percases ( x1 = B or x1 = C ) by A7, A9, A19, TARSKI:def_2; case x1 = B ; ::_thesis: u in {(h . B),(h . C)} hence u in {(h . B),(h . C)} by A20, TARSKI:def_2; ::_thesis: verum end; case x1 = C ; ::_thesis: u in {(h . B),(h . C)} hence u in {(h . B),(h . C)} by A20, TARSKI:def_2; ::_thesis: verum end; end; end; hence u in {(h . B),(h . C)} ; ::_thesis: verum end; for y being set st y in rng h holds m in y proof let y be set ; ::_thesis: ( y in rng h implies m in y ) assume A21: y in rng h ; ::_thesis: m in y now__::_thesis:_(_(_y_=_h_._B_&_m_in_y_)_or_(_y_=_h_._C_&_m_in_y_)_) percases ( y = h . B or y = h . C ) by A18, A21, TARSKI:def_2; case y = h . B ; ::_thesis: m in y hence m in y by A17, XBOOLE_0:def_4; ::_thesis: verum end; case y = h . C ; ::_thesis: m in y hence m in y by A17, XBOOLE_0:def_4; ::_thesis: verum end; end; end; hence m in y ; ::_thesis: verum end; then m in meet (rng h) by A15, SETFAM_1:def_1; hence m in x by A10, A12, A15, SETFAM_1:def_9; ::_thesis: verum end; C in G \ {A} by A7, TARSKI:def_2; then A22: h . C in C by A11; B in G \ {A} by A7, TARSKI:def_2; then A23: h . B in B by A11; C in dom h by A7, A9, TARSKI:def_2; then A24: h . C in rng h by FUNCT_1:def_3; x c= (h . B) /\ (h . C) proof let m be set ; :: according to TARSKI:def_3 ::_thesis: ( not m in x or m in (h . B) /\ (h . C) ) assume m in x ; ::_thesis: m in (h . B) /\ (h . C) then m in meet (rng h) by A10, A12, A15, SETFAM_1:def_9; then ( m in h . B & m in h . C ) by A15, A24, SETFAM_1:def_1; hence m in (h . B) /\ (h . C) by XBOOLE_0:def_4; ::_thesis: verum end; then (h . B) /\ (h . C) = x by A16, XBOOLE_0:def_10; then x in INTERSECTION (B,C) by A23, A22, SETFAM_1:def_5; then x in (INTERSECTION (B,C)) \ {{}} by A14, XBOOLE_0:def_5; hence x in B '/\' C by PARTIT1:def_4; ::_thesis: verum end; A25: B '/\' C c= '/\' (G \ {A}) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in B '/\' C or x in '/\' (G \ {A}) ) assume A26: x in B '/\' C ; ::_thesis: x in '/\' (G \ {A}) then A27: x <> {} by EQREL_1:def_4; x in (INTERSECTION (B,C)) \ {{}} by A26, PARTIT1:def_4; then consider a, b being set such that A28: a in B and A29: b in C and A30: x = a /\ b by SETFAM_1:def_5; set h0 = (B,C) --> (a,b); A31: dom ((B,C) --> (a,b)) = G \ {A} by A7, FUNCT_4:62; A32: rng ((B,C) --> (a,b)) = {a,b} by A5, FUNCT_4:64; rng ((B,C) --> (a,b)) c= bool Y proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng ((B,C) --> (a,b)) or y in bool Y ) assume A33: y in rng ((B,C) --> (a,b)) ; ::_thesis: y in bool Y now__::_thesis:_(_(_y_=_a_&_y_in_bool_Y_)_or_(_y_=_b_&_y_in_bool_Y_)_) percases ( y = a or y = b ) by A32, A33, TARSKI:def_2; case y = a ; ::_thesis: y in bool Y hence y in bool Y by A28; ::_thesis: verum end; case y = b ; ::_thesis: y in bool Y hence y in bool Y by A29; ::_thesis: verum end; end; end; hence y in bool Y ; ::_thesis: verum end; then reconsider F = rng ((B,C) --> (a,b)) as Subset-Family of Y ; A34: x c= Intersect F proof let u be set ; :: according to TARSKI:def_3 ::_thesis: ( not u in x or u in Intersect F ) assume A35: u in x ; ::_thesis: u in Intersect F for y being set st y in F holds u in y proof let y be set ; ::_thesis: ( y in F implies u in y ) assume A36: y in F ; ::_thesis: u in y now__::_thesis:_(_(_y_=_a_&_u_in_y_)_or_(_y_=_b_&_u_in_y_)_) percases ( y = a or y = b ) by A32, A36, TARSKI:def_2; case y = a ; ::_thesis: u in y hence u in y by A30, A35, XBOOLE_0:def_4; ::_thesis: verum end; case y = b ; ::_thesis: u in y hence u in y by A30, A35, XBOOLE_0:def_4; ::_thesis: verum end; end; end; hence u in y ; ::_thesis: verum end; then u in meet F by A32, SETFAM_1:def_1; hence u in Intersect F by A32, SETFAM_1:def_9; ::_thesis: verum end; A37: for d being set st d in G \ {A} holds ((B,C) --> (a,b)) . d in d proof let d be set ; ::_thesis: ( d in G \ {A} implies ((B,C) --> (a,b)) . d in d ) assume A38: d in G \ {A} ; ::_thesis: ((B,C) --> (a,b)) . d in d now__::_thesis:_(_(_d_=_B_&_((B,C)_-->_(a,b))_._d_in_d_)_or_(_d_=_C_&_((B,C)_-->_(a,b))_._d_in_d_)_) percases ( d = B or d = C ) by A7, A38, TARSKI:def_2; case d = B ; ::_thesis: ((B,C) --> (a,b)) . d in d hence ((B,C) --> (a,b)) . d in d by A5, A28, FUNCT_4:63; ::_thesis: verum end; case d = C ; ::_thesis: ((B,C) --> (a,b)) . d in d hence ((B,C) --> (a,b)) . d in d by A29, FUNCT_4:63; ::_thesis: verum end; end; end; hence ((B,C) --> (a,b)) . d in d ; ::_thesis: verum end; Intersect F c= x proof let u be set ; :: according to TARSKI:def_3 ::_thesis: ( not u in Intersect F or u in x ) assume A39: u in Intersect F ; ::_thesis: u in x A40: Intersect F = meet F by A32, SETFAM_1:def_9; b in F by A32, TARSKI:def_2; then A41: u in b by A39, A40, SETFAM_1:def_1; a in F by A32, TARSKI:def_2; then u in a by A39, A40, SETFAM_1:def_1; hence u in x by A30, A41, XBOOLE_0:def_4; ::_thesis: verum end; then x = Intersect F by A34, XBOOLE_0:def_10; hence x in '/\' (G \ {A}) by A31, A37, A27, BVFUNC_2:def_1; ::_thesis: verum end; CompF (A,G) = '/\' (G \ {A}) by BVFUNC_2:def_7; hence CompF (A,G) = B '/\' C by A25, A8, XBOOLE_0:def_10; ::_thesis: verum end; end; end; theorem Th5: :: BVFUNC14:5 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C being a_partition of Y st G = {A,B,C} & A <> B & B <> C holds CompF (B,G) = C '/\' A proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C being a_partition of Y st G = {A,B,C} & A <> B & B <> C holds CompF (B,G) = C '/\' A let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C being a_partition of Y st G = {A,B,C} & A <> B & B <> C holds CompF (B,G) = C '/\' A let A, B, C be a_partition of Y; ::_thesis: ( G = {A,B,C} & A <> B & B <> C implies CompF (B,G) = C '/\' A ) {A,B,C} = {B,C,A} by ENUMSET1:59; hence ( G = {A,B,C} & A <> B & B <> C implies CompF (B,G) = C '/\' A ) by Th4; ::_thesis: verum end; theorem :: BVFUNC14:6 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C being a_partition of Y st G = {A,B,C} & B <> C & C <> A holds CompF (C,G) = A '/\' B proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C being a_partition of Y st G = {A,B,C} & B <> C & C <> A holds CompF (C,G) = A '/\' B let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C being a_partition of Y st G = {A,B,C} & B <> C & C <> A holds CompF (C,G) = A '/\' B let A, B, C be a_partition of Y; ::_thesis: ( G = {A,B,C} & B <> C & C <> A implies CompF (C,G) = A '/\' B ) {A,B,C} = {C,A,B} by ENUMSET1:59; hence ( G = {A,B,C} & B <> C & C <> A implies CompF (C,G) = A '/\' B ) by Th4; ::_thesis: verum end; theorem Th7: :: BVFUNC14:7 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D being a_partition of Y st G = {A,B,C,D} & A <> B & A <> C & A <> D holds CompF (A,G) = (B '/\' C) '/\' D proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C, D being a_partition of Y st G = {A,B,C,D} & A <> B & A <> C & A <> D holds CompF (A,G) = (B '/\' C) '/\' D let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C, D being a_partition of Y st G = {A,B,C,D} & A <> B & A <> C & A <> D holds CompF (A,G) = (B '/\' C) '/\' D let A, B, C, D be a_partition of Y; ::_thesis: ( G = {A,B,C,D} & A <> B & A <> C & A <> D implies CompF (A,G) = (B '/\' C) '/\' D ) assume that A1: G = {A,B,C,D} and A2: A <> B and A3: A <> C and A4: A <> D ; ::_thesis: CompF (A,G) = (B '/\' C) '/\' D percases ( B = C or B = D or C = D or ( B <> C & B <> D & C <> D ) ) ; supposeA5: B = C ; ::_thesis: CompF (A,G) = (B '/\' C) '/\' D then G = {B,B,A,D} by A1, ENUMSET1:71 .= {B,A,D} by ENUMSET1:31 .= {A,B,D} by ENUMSET1:58 ; hence CompF (A,G) = B '/\' D by A2, A4, Th4 .= (B '/\' C) '/\' D by A5, PARTIT1:13 ; ::_thesis: verum end; supposeA6: B = D ; ::_thesis: CompF (A,G) = (B '/\' C) '/\' D then G = {B,B,A,C} by A1, ENUMSET1:69 .= {B,A,C} by ENUMSET1:31 .= {A,B,C} by ENUMSET1:58 ; hence CompF (A,G) = B '/\' C by A2, A3, Th4 .= (B '/\' D) '/\' C by A6, PARTIT1:13 .= (B '/\' C) '/\' D by PARTIT1:14 ; ::_thesis: verum end; supposeA7: C = D ; ::_thesis: CompF (A,G) = (B '/\' C) '/\' D then G = {C,C,A,B} by A1, ENUMSET1:73 .= {C,A,B} by ENUMSET1:31 .= {A,B,C} by ENUMSET1:59 ; hence CompF (A,G) = B '/\' C by A2, A3, Th4 .= B '/\' (C '/\' D) by A7, PARTIT1:13 .= (B '/\' C) '/\' D by PARTIT1:14 ; ::_thesis: verum end; supposeA8: ( B <> C & B <> D & C <> D ) ; ::_thesis: CompF (A,G) = (B '/\' C) '/\' D G \ {A} = ({A} \/ {B,C,D}) \ {A} by A1, ENUMSET1:4; then A9: G \ {A} = ({A} \ {A}) \/ ({B,C,D} \ {A}) by XBOOLE_1:42; A10: not B in {A} by A2, TARSKI:def_1; A11: ( not C in {A} & not D in {A} ) by A3, A4, TARSKI:def_1; {B,C,D} \ {A} = ({B} \/ {C,D}) \ {A} by ENUMSET1:2 .= ({B} \ {A}) \/ ({C,D} \ {A}) by XBOOLE_1:42 .= ({B} \ {A}) \/ {C,D} by A11, ZFMISC_1:63 .= {B} \/ {C,D} by A10, ZFMISC_1:59 .= {B,C,D} by ENUMSET1:2 ; then A12: G \ {A} = {} \/ {B,C,D} by A9, XBOOLE_1:37 .= {B,C,D} ; A13: (B '/\' C) '/\' D c= '/\' (G \ {A}) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (B '/\' C) '/\' D or x in '/\' (G \ {A}) ) assume A14: x in (B '/\' C) '/\' D ; ::_thesis: x in '/\' (G \ {A}) then A15: x <> {} by EQREL_1:def_4; x in (INTERSECTION ((B '/\' C),D)) \ {{}} by A14, PARTIT1:def_4; then consider a, d being set such that A16: a in B '/\' C and A17: d in D and A18: x = a /\ d by SETFAM_1:def_5; a in (INTERSECTION (B,C)) \ {{}} by A16, PARTIT1:def_4; then consider b, c being set such that A19: b in B and A20: c in C and A21: a = b /\ c by SETFAM_1:def_5; set h = ((B .--> b) +* (C .--> c)) +* (D .--> d); A22: (((B .--> b) +* (C .--> c)) +* (D .--> d)) . D = d by FUNCT_7:94; A23: (((B .--> b) +* (C .--> c)) +* (D .--> d)) . C = c by A8, Lm2; A24: rng (((B .--> b) +* (C .--> c)) +* (D .--> d)) = {((((B .--> b) +* (C .--> c)) +* (D .--> d)) . B),((((B .--> b) +* (C .--> c)) +* (D .--> d)) . C),((((B .--> b) +* (C .--> c)) +* (D .--> d)) . D)} by Lm4 .= {((((B .--> b) +* (C .--> c)) +* (D .--> d)) . D),((((B .--> b) +* (C .--> c)) +* (D .--> d)) . B),((((B .--> b) +* (C .--> c)) +* (D .--> d)) . C)} by ENUMSET1:59 ; A25: (((B .--> b) +* (C .--> c)) +* (D .--> d)) . B = b by A8, Lm3; rng (((B .--> b) +* (C .--> c)) +* (D .--> d)) c= bool Y proof let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in rng (((B .--> b) +* (C .--> c)) +* (D .--> d)) or t in bool Y ) assume A26: t in rng (((B .--> b) +* (C .--> c)) +* (D .--> d)) ; ::_thesis: t in bool Y now__::_thesis:_(_(_t_=_(((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_._D_&_t_in_bool_Y_)_or_(_t_=_(((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_._B_&_t_in_bool_Y_)_or_(_t_=_(((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_._C_&_t_in_bool_Y_)_) percases ( t = (((B .--> b) +* (C .--> c)) +* (D .--> d)) . D or t = (((B .--> b) +* (C .--> c)) +* (D .--> d)) . B or t = (((B .--> b) +* (C .--> c)) +* (D .--> d)) . C ) by A24, A26, ENUMSET1:def_1; case t = (((B .--> b) +* (C .--> c)) +* (D .--> d)) . D ; ::_thesis: t in bool Y hence t in bool Y by A17, A22; ::_thesis: verum end; case t = (((B .--> b) +* (C .--> c)) +* (D .--> d)) . B ; ::_thesis: t in bool Y hence t in bool Y by A19, A25; ::_thesis: verum end; case t = (((B .--> b) +* (C .--> c)) +* (D .--> d)) . C ; ::_thesis: t in bool Y hence t in bool Y by A20, A23; ::_thesis: verum end; end; end; hence t in bool Y ; ::_thesis: verum end; then reconsider F = rng (((B .--> b) +* (C .--> c)) +* (D .--> d)) as Subset-Family of Y ; A27: x c= Intersect F proof let u be set ; :: according to TARSKI:def_3 ::_thesis: ( not u in x or u in Intersect F ) assume A28: u in x ; ::_thesis: u in Intersect F for y being set st y in F holds u in y proof let y be set ; ::_thesis: ( y in F implies u in y ) assume A29: y in F ; ::_thesis: u in y now__::_thesis:_(_(_y_=_(((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_._D_&_u_in_y_)_or_(_y_=_(((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_._B_&_u_in_y_)_or_(_y_=_(((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_._C_&_u_in_y_)_) percases ( y = (((B .--> b) +* (C .--> c)) +* (D .--> d)) . D or y = (((B .--> b) +* (C .--> c)) +* (D .--> d)) . B or y = (((B .--> b) +* (C .--> c)) +* (D .--> d)) . C ) by A24, A29, ENUMSET1:def_1; case y = (((B .--> b) +* (C .--> c)) +* (D .--> d)) . D ; ::_thesis: u in y hence u in y by A18, A22, A28, XBOOLE_0:def_4; ::_thesis: verum end; caseA30: y = (((B .--> b) +* (C .--> c)) +* (D .--> d)) . B ; ::_thesis: u in y u in b /\ (c /\ d) by A18, A21, A28, XBOOLE_1:16; hence u in y by A25, A30, XBOOLE_0:def_4; ::_thesis: verum end; caseA31: y = (((B .--> b) +* (C .--> c)) +* (D .--> d)) . C ; ::_thesis: u in y u in c /\ (b /\ d) by A18, A21, A28, XBOOLE_1:16; hence u in y by A23, A31, XBOOLE_0:def_4; ::_thesis: verum end; end; end; hence u in y ; ::_thesis: verum end; then u in meet F by A24, SETFAM_1:def_1; hence u in Intersect F by A24, SETFAM_1:def_9; ::_thesis: verum end; A32: for p being set st p in G \ {A} holds (((B .--> b) +* (C .--> c)) +* (D .--> d)) . p in p proof let p be set ; ::_thesis: ( p in G \ {A} implies (((B .--> b) +* (C .--> c)) +* (D .--> d)) . p in p ) assume A33: p in G \ {A} ; ::_thesis: (((B .--> b) +* (C .--> c)) +* (D .--> d)) . p in p now__::_thesis:_(_(_p_=_D_&_(((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_._p_in_p_)_or_(_p_=_B_&_(((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_._p_in_p_)_or_(_p_=_C_&_(((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_._p_in_p_)_) percases ( p = D or p = B or p = C ) by A12, A33, ENUMSET1:def_1; case p = D ; ::_thesis: (((B .--> b) +* (C .--> c)) +* (D .--> d)) . p in p hence (((B .--> b) +* (C .--> c)) +* (D .--> d)) . p in p by A17, FUNCT_7:94; ::_thesis: verum end; case p = B ; ::_thesis: (((B .--> b) +* (C .--> c)) +* (D .--> d)) . p in p hence (((B .--> b) +* (C .--> c)) +* (D .--> d)) . p in p by A8, A19, Lm3; ::_thesis: verum end; case p = C ; ::_thesis: (((B .--> b) +* (C .--> c)) +* (D .--> d)) . p in p hence (((B .--> b) +* (C .--> c)) +* (D .--> d)) . p in p by A8, A20, Lm2; ::_thesis: verum end; end; end; hence (((B .--> b) +* (C .--> c)) +* (D .--> d)) . p in p ; ::_thesis: verum end; A34: dom (((B .--> b) +* (C .--> c)) +* (D .--> d)) = {B,C,D} by Lm1; then D in dom (((B .--> b) +* (C .--> c)) +* (D .--> d)) by ENUMSET1:def_1; then A35: rng (((B .--> b) +* (C .--> c)) +* (D .--> d)) <> {} by FUNCT_1:3; Intersect F c= x proof let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in Intersect F or t in x ) assume t in Intersect F ; ::_thesis: t in x then A36: t in meet (rng (((B .--> b) +* (C .--> c)) +* (D .--> d))) by A35, SETFAM_1:def_9; (((B .--> b) +* (C .--> c)) +* (D .--> d)) . D in rng (((B .--> b) +* (C .--> c)) +* (D .--> d)) by A24, ENUMSET1:def_1; then A37: t in (((B .--> b) +* (C .--> c)) +* (D .--> d)) . D by A36, SETFAM_1:def_1; (((B .--> b) +* (C .--> c)) +* (D .--> d)) . C in rng (((B .--> b) +* (C .--> c)) +* (D .--> d)) by A24, ENUMSET1:def_1; then A38: t in (((B .--> b) +* (C .--> c)) +* (D .--> d)) . C by A36, SETFAM_1:def_1; (((B .--> b) +* (C .--> c)) +* (D .--> d)) . B in rng (((B .--> b) +* (C .--> c)) +* (D .--> d)) by A24, ENUMSET1:def_1; then t in (((B .--> b) +* (C .--> c)) +* (D .--> d)) . B by A36, SETFAM_1:def_1; then t in b /\ c by A25, A23, A38, XBOOLE_0:def_4; hence t in x by A18, A21, A22, A37, XBOOLE_0:def_4; ::_thesis: verum end; then x = Intersect F by A27, XBOOLE_0:def_10; hence x in '/\' (G \ {A}) by A12, A34, A32, A15, BVFUNC_2:def_1; ::_thesis: verum end; '/\' (G \ {A}) c= (B '/\' C) '/\' D proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in '/\' (G \ {A}) or x in (B '/\' C) '/\' D ) assume x in '/\' (G \ {A}) ; ::_thesis: x in (B '/\' C) '/\' D then consider h being Function, F being Subset-Family of Y such that A39: dom h = G \ {A} and A40: rng h = F and A41: for d being set st d in G \ {A} holds h . d in d and A42: x = Intersect F and A43: x <> {} by BVFUNC_2:def_1; D in dom h by A12, A39, ENUMSET1:def_1; then A44: h . D in rng h by FUNCT_1:def_3; set m = (h . B) /\ (h . C); B in dom h by A12, A39, ENUMSET1:def_1; then A45: h . B in rng h by FUNCT_1:def_3; C in dom h by A12, A39, ENUMSET1:def_1; then A46: h . C in rng h by FUNCT_1:def_3; A47: x c= ((h . B) /\ (h . C)) /\ (h . D) proof let m be set ; :: according to TARSKI:def_3 ::_thesis: ( not m in x or m in ((h . B) /\ (h . C)) /\ (h . D) ) assume m in x ; ::_thesis: m in ((h . B) /\ (h . C)) /\ (h . D) then A48: m in meet (rng h) by A40, A42, A45, SETFAM_1:def_9; then ( m in h . B & m in h . C ) by A45, A46, SETFAM_1:def_1; then A49: m in (h . B) /\ (h . C) by XBOOLE_0:def_4; m in h . D by A44, A48, SETFAM_1:def_1; hence m in ((h . B) /\ (h . C)) /\ (h . D) by A49, XBOOLE_0:def_4; ::_thesis: verum end; then (h . B) /\ (h . C) <> {} by A43; then A50: not (h . B) /\ (h . C) in {{}} by TARSKI:def_1; D in G \ {A} by A12, ENUMSET1:def_1; then A51: h . D in D by A41; A52: not x in {{}} by A43, TARSKI:def_1; C in G \ {A} by A12, ENUMSET1:def_1; then A53: h . C in C by A41; B in G \ {A} by A12, ENUMSET1:def_1; then h . B in B by A41; then (h . B) /\ (h . C) in INTERSECTION (B,C) by A53, SETFAM_1:def_5; then (h . B) /\ (h . C) in (INTERSECTION (B,C)) \ {{}} by A50, XBOOLE_0:def_5; then A54: (h . B) /\ (h . C) in B '/\' C by PARTIT1:def_4; ((h . B) /\ (h . C)) /\ (h . D) c= x proof let m be set ; :: according to TARSKI:def_3 ::_thesis: ( not m in ((h . B) /\ (h . C)) /\ (h . D) or m in x ) assume A55: m in ((h . B) /\ (h . C)) /\ (h . D) ; ::_thesis: m in x then A56: m in (h . B) /\ (h . C) by XBOOLE_0:def_4; A57: rng h c= {(h . B),(h . C),(h . D)} proof let u be set ; :: according to TARSKI:def_3 ::_thesis: ( not u in rng h or u in {(h . B),(h . C),(h . D)} ) assume u in rng h ; ::_thesis: u in {(h . B),(h . C),(h . D)} then consider x1 being set such that A58: x1 in dom h and A59: u = h . x1 by FUNCT_1:def_3; now__::_thesis:_(_(_x1_=_B_&_u_in_{(h_._B),(h_._C),(h_._D)}_)_or_(_x1_=_C_&_u_in_{(h_._B),(h_._C),(h_._D)}_)_or_(_x1_=_D_&_u_in_{(h_._B),(h_._C),(h_._D)}_)_) percases ( x1 = B or x1 = C or x1 = D ) by A12, A39, A58, ENUMSET1:def_1; case x1 = B ; ::_thesis: u in {(h . B),(h . C),(h . D)} hence u in {(h . B),(h . C),(h . D)} by A59, ENUMSET1:def_1; ::_thesis: verum end; case x1 = C ; ::_thesis: u in {(h . B),(h . C),(h . D)} hence u in {(h . B),(h . C),(h . D)} by A59, ENUMSET1:def_1; ::_thesis: verum end; case x1 = D ; ::_thesis: u in {(h . B),(h . C),(h . D)} hence u in {(h . B),(h . C),(h . D)} by A59, ENUMSET1:def_1; ::_thesis: verum end; end; end; hence u in {(h . B),(h . C),(h . D)} ; ::_thesis: verum end; for y being set st y in rng h holds m in y proof let y be set ; ::_thesis: ( y in rng h implies m in y ) assume A60: y in rng h ; ::_thesis: m in y now__::_thesis:_(_(_y_=_h_._B_&_m_in_y_)_or_(_y_=_h_._C_&_m_in_y_)_or_(_y_=_h_._D_&_m_in_y_)_) percases ( y = h . B or y = h . C or y = h . D ) by A57, A60, ENUMSET1:def_1; case y = h . B ; ::_thesis: m in y hence m in y by A56, XBOOLE_0:def_4; ::_thesis: verum end; case y = h . C ; ::_thesis: m in y hence m in y by A56, XBOOLE_0:def_4; ::_thesis: verum end; case y = h . D ; ::_thesis: m in y hence m in y by A55, XBOOLE_0:def_4; ::_thesis: verum end; end; end; hence m in y ; ::_thesis: verum end; then m in meet (rng h) by A45, SETFAM_1:def_1; hence m in x by A40, A42, A45, SETFAM_1:def_9; ::_thesis: verum end; then ((h . B) /\ (h . C)) /\ (h . D) = x by A47, XBOOLE_0:def_10; then x in INTERSECTION ((B '/\' C),D) by A51, A54, SETFAM_1:def_5; then x in (INTERSECTION ((B '/\' C),D)) \ {{}} by A52, XBOOLE_0:def_5; hence x in (B '/\' C) '/\' D by PARTIT1:def_4; ::_thesis: verum end; then '/\' (G \ {A}) = (B '/\' C) '/\' D by A13, XBOOLE_0:def_10; hence CompF (A,G) = (B '/\' C) '/\' D by BVFUNC_2:def_7; ::_thesis: verum end; end; end; theorem Th8: :: BVFUNC14:8 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D being a_partition of Y st G = {A,B,C,D} & A <> B & B <> C & B <> D holds CompF (B,G) = (A '/\' C) '/\' D proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C, D being a_partition of Y st G = {A,B,C,D} & A <> B & B <> C & B <> D holds CompF (B,G) = (A '/\' C) '/\' D let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C, D being a_partition of Y st G = {A,B,C,D} & A <> B & B <> C & B <> D holds CompF (B,G) = (A '/\' C) '/\' D let A, B, C, D be a_partition of Y; ::_thesis: ( G = {A,B,C,D} & A <> B & B <> C & B <> D implies CompF (B,G) = (A '/\' C) '/\' D ) {A,B,C,D} = {B,A,C,D} by ENUMSET1:65; hence ( G = {A,B,C,D} & A <> B & B <> C & B <> D implies CompF (B,G) = (A '/\' C) '/\' D ) by Th7; ::_thesis: verum end; theorem :: BVFUNC14:9 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D being a_partition of Y st G = {A,B,C,D} & A <> C & B <> C & C <> D holds CompF (C,G) = (A '/\' B) '/\' D proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C, D being a_partition of Y st G = {A,B,C,D} & A <> C & B <> C & C <> D holds CompF (C,G) = (A '/\' B) '/\' D let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C, D being a_partition of Y st G = {A,B,C,D} & A <> C & B <> C & C <> D holds CompF (C,G) = (A '/\' B) '/\' D let A, B, C, D be a_partition of Y; ::_thesis: ( G = {A,B,C,D} & A <> C & B <> C & C <> D implies CompF (C,G) = (A '/\' B) '/\' D ) {A,B,C,D} = {C,A,B,D} by ENUMSET1:67; hence ( G = {A,B,C,D} & A <> C & B <> C & C <> D implies CompF (C,G) = (A '/\' B) '/\' D ) by Th7; ::_thesis: verum end; theorem :: BVFUNC14:10 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D being a_partition of Y st G = {A,B,C,D} & A <> D & B <> D & C <> D holds CompF (D,G) = (A '/\' C) '/\' B proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C, D being a_partition of Y st G = {A,B,C,D} & A <> D & B <> D & C <> D holds CompF (D,G) = (A '/\' C) '/\' B let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C, D being a_partition of Y st G = {A,B,C,D} & A <> D & B <> D & C <> D holds CompF (D,G) = (A '/\' C) '/\' B let A, B, C, D be a_partition of Y; ::_thesis: ( G = {A,B,C,D} & A <> D & B <> D & C <> D implies CompF (D,G) = (A '/\' C) '/\' B ) {A,B,C,D} = {D,A,C,B} by ENUMSET1:70; hence ( G = {A,B,C,D} & A <> D & B <> D & C <> D implies CompF (D,G) = (A '/\' C) '/\' B ) by Th7; ::_thesis: verum end; theorem :: BVFUNC14:11 for B, C, D, b, c, d being set holds dom (((B .--> b) +* (C .--> c)) +* (D .--> d)) = {B,C,D} by Lm1; theorem :: BVFUNC14:12 for f being Function for C, D, c, d being set st C <> D holds ((f +* (C .--> c)) +* (D .--> d)) . C = c by Lm2; theorem :: BVFUNC14:13 for B, C, D, b, c, d being set st B <> C & D <> B holds (((B .--> b) +* (C .--> c)) +* (D .--> d)) . B = b by Lm3; theorem :: BVFUNC14:14 for B, C, D, b, c, d being set for h being Function st h = ((B .--> b) +* (C .--> c)) +* (D .--> d) holds rng h = {(h . B),(h . C),(h . D)} by Lm4; theorem Th15: :: BVFUNC14:15 for Y being non empty set for A, B, C, D being a_partition of Y for h being Function for A9, B9, C9, D9 being set st A <> B & A <> C & A <> D & B <> C & B <> D & C <> D & h = (((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (A .--> A9) holds ( h . B = B9 & h . C = C9 & h . D = D9 ) proof let Y be non empty set ; ::_thesis: for A, B, C, D being a_partition of Y for h being Function for A9, B9, C9, D9 being set st A <> B & A <> C & A <> D & B <> C & B <> D & C <> D & h = (((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (A .--> A9) holds ( h . B = B9 & h . C = C9 & h . D = D9 ) let A, B, C, D be a_partition of Y; ::_thesis: for h being Function for A9, B9, C9, D9 being set st A <> B & A <> C & A <> D & B <> C & B <> D & C <> D & h = (((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (A .--> A9) holds ( h . B = B9 & h . C = C9 & h . D = D9 ) let h be Function; ::_thesis: for A9, B9, C9, D9 being set st A <> B & A <> C & A <> D & B <> C & B <> D & C <> D & h = (((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (A .--> A9) holds ( h . B = B9 & h . C = C9 & h . D = D9 ) let A9, B9, C9, D9 be set ; ::_thesis: ( A <> B & A <> C & A <> D & B <> C & B <> D & C <> D & h = (((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (A .--> A9) implies ( h . B = B9 & h . C = C9 & h . D = D9 ) ) assume that A1: A <> B and A2: A <> C and A3: A <> D and A4: B <> C and A5: B <> D and A6: C <> D and A7: h = (((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (A .--> A9) ; ::_thesis: ( h . B = B9 & h . C = C9 & h . D = D9 ) A8: dom (A .--> A9) = {A} by FUNCOP_1:13; then not D in dom (A .--> A9) by A3, TARSKI:def_1; then A9: h . D = (((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) . D by A7, FUNCT_4:11; not C in dom (A .--> A9) by A2, A8, TARSKI:def_1; then A10: h . C = (((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) . C by A7, FUNCT_4:11; A11: dom (D .--> D9) = {D} by FUNCOP_1:13; then not C in dom (D .--> D9) by A6, TARSKI:def_1; then A12: h . C = ((B .--> B9) +* (C .--> C9)) . C by A10, FUNCT_4:11; not B in dom (A .--> A9) by A1, A8, TARSKI:def_1; then A13: h . B = (((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) . B by A7, FUNCT_4:11; not B in dom (D .--> D9) by A5, A11, TARSKI:def_1; then A14: h . B = ((B .--> B9) +* (C .--> C9)) . B by A13, FUNCT_4:11; A15: dom (C .--> C9) = {C} by FUNCOP_1:13; then not B in dom (C .--> C9) by A4, TARSKI:def_1; then h . B = (B .--> B9) . B by A14, FUNCT_4:11; hence h . B = B9 by FUNCOP_1:72; ::_thesis: ( h . C = C9 & h . D = D9 ) C in dom (C .--> C9) by A15, TARSKI:def_1; then h . C = (C .--> C9) . C by A12, FUNCT_4:13; hence h . C = C9 by FUNCOP_1:72; ::_thesis: h . D = D9 D in dom (D .--> D9) by A11, TARSKI:def_1; then h . D = (D .--> D9) . D by A9, FUNCT_4:13; hence h . D = D9 by FUNCOP_1:72; ::_thesis: verum end; theorem Th16: :: BVFUNC14:16 for A, B, C, D being set for h being Function for A9, B9, C9, D9 being set st h = (((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (A .--> A9) holds dom h = {A,B,C,D} proof let A, B, C, D be set ; ::_thesis: for h being Function for A9, B9, C9, D9 being set st h = (((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (A .--> A9) holds dom h = {A,B,C,D} let h be Function; ::_thesis: for A9, B9, C9, D9 being set st h = (((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (A .--> A9) holds dom h = {A,B,C,D} let A9, B9, C9, D9 be set ; ::_thesis: ( h = (((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (A .--> A9) implies dom h = {A,B,C,D} ) assume A1: h = (((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (A .--> A9) ; ::_thesis: dom h = {A,B,C,D} dom ((B .--> B9) +* (C .--> C9)) = (dom (B .--> B9)) \/ (dom (C .--> C9)) by FUNCT_4:def_1; then A2: dom (((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) = ((dom (B .--> B9)) \/ (dom (C .--> C9))) \/ (dom (D .--> D9)) by FUNCT_4:def_1; dom (B .--> B9) = {B} by FUNCOP_1:13; then dom ((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (A .--> A9)) = (({B} \/ (dom (C .--> C9))) \/ (dom (D .--> D9))) \/ (dom (A .--> A9)) by A2, FUNCT_4:def_1 .= (({B} \/ {C}) \/ (dom (D .--> D9))) \/ (dom (A .--> A9)) by FUNCOP_1:13 .= (({B} \/ {C}) \/ {D}) \/ (dom (A .--> A9)) by FUNCOP_1:13 .= {A} \/ (({B} \/ {C}) \/ {D}) by FUNCOP_1:13 .= {A} \/ ({B,C} \/ {D}) by ENUMSET1:1 .= {A} \/ {B,C,D} by ENUMSET1:3 .= {A,B,C,D} by ENUMSET1:4 ; hence dom h = {A,B,C,D} by A1; ::_thesis: verum end; theorem Th17: :: BVFUNC14:17 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D being a_partition of Y for h being Function for A9, B9, C9, D9 being set st G = {A,B,C,D} & h = (((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (A .--> A9) holds rng h = {(h . A),(h . B),(h . C),(h . D)} proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C, D being a_partition of Y for h being Function for A9, B9, C9, D9 being set st G = {A,B,C,D} & h = (((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (A .--> A9) holds rng h = {(h . A),(h . B),(h . C),(h . D)} let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C, D being a_partition of Y for h being Function for A9, B9, C9, D9 being set st G = {A,B,C,D} & h = (((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (A .--> A9) holds rng h = {(h . A),(h . B),(h . C),(h . D)} let A, B, C, D be a_partition of Y; ::_thesis: for h being Function for A9, B9, C9, D9 being set st G = {A,B,C,D} & h = (((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (A .--> A9) holds rng h = {(h . A),(h . B),(h . C),(h . D)} let h be Function; ::_thesis: for A9, B9, C9, D9 being set st G = {A,B,C,D} & h = (((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (A .--> A9) holds rng h = {(h . A),(h . B),(h . C),(h . D)} let A9, B9, C9, D9 be set ; ::_thesis: ( G = {A,B,C,D} & h = (((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (A .--> A9) implies rng h = {(h . A),(h . B),(h . C),(h . D)} ) assume that A1: G = {A,B,C,D} and A2: h = (((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (A .--> A9) ; ::_thesis: rng h = {(h . A),(h . B),(h . C),(h . D)} A3: dom h = G by A1, A2, Th16; then A4: B in dom h by A1, ENUMSET1:def_2; A5: rng h c= {(h . A),(h . B),(h . C),(h . D)} proof let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in rng h or t in {(h . A),(h . B),(h . C),(h . D)} ) assume t in rng h ; ::_thesis: t in {(h . A),(h . B),(h . C),(h . D)} then consider x1 being set such that A6: x1 in dom h and A7: t = h . x1 by FUNCT_1:def_3; now__::_thesis:_(_(_x1_=_A_&_t_in_{(h_._A),(h_._B),(h_._C),(h_._D)}_)_or_(_x1_=_B_&_t_in_{(h_._A),(h_._B),(h_._C),(h_._D)}_)_or_(_x1_=_C_&_t_in_{(h_._A),(h_._B),(h_._C),(h_._D)}_)_or_(_x1_=_D_&_t_in_{(h_._A),(h_._B),(h_._C),(h_._D)}_)_) percases ( x1 = A or x1 = B or x1 = C or x1 = D ) by A1, A3, A6, ENUMSET1:def_2; case x1 = A ; ::_thesis: t in {(h . A),(h . B),(h . C),(h . D)} hence t in {(h . A),(h . B),(h . C),(h . D)} by A7, ENUMSET1:def_2; ::_thesis: verum end; case x1 = B ; ::_thesis: t in {(h . A),(h . B),(h . C),(h . D)} hence t in {(h . A),(h . B),(h . C),(h . D)} by A7, ENUMSET1:def_2; ::_thesis: verum end; case x1 = C ; ::_thesis: t in {(h . A),(h . B),(h . C),(h . D)} hence t in {(h . A),(h . B),(h . C),(h . D)} by A7, ENUMSET1:def_2; ::_thesis: verum end; case x1 = D ; ::_thesis: t in {(h . A),(h . B),(h . C),(h . D)} hence t in {(h . A),(h . B),(h . C),(h . D)} by A7, ENUMSET1:def_2; ::_thesis: verum end; end; end; hence t in {(h . A),(h . B),(h . C),(h . D)} ; ::_thesis: verum end; A8: D in dom h by A1, A3, ENUMSET1:def_2; A9: C in dom h by A1, A3, ENUMSET1:def_2; A10: A in dom h by A1, A3, ENUMSET1:def_2; {(h . A),(h . B),(h . C),(h . D)} c= rng h proof let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in {(h . A),(h . B),(h . C),(h . D)} or t in rng h ) assume A11: t in {(h . A),(h . B),(h . C),(h . D)} ; ::_thesis: t in rng h percases ( t = h . A or t = h . B or t = h . C or t = h . D ) by A11, ENUMSET1:def_2; suppose t = h . A ; ::_thesis: t in rng h hence t in rng h by A10, FUNCT_1:def_3; ::_thesis: verum end; suppose t = h . B ; ::_thesis: t in rng h hence t in rng h by A4, FUNCT_1:def_3; ::_thesis: verum end; suppose t = h . C ; ::_thesis: t in rng h hence t in rng h by A9, FUNCT_1:def_3; ::_thesis: verum end; suppose t = h . D ; ::_thesis: t in rng h hence t in rng h by A8, FUNCT_1:def_3; ::_thesis: verum end; end; end; hence rng h = {(h . A),(h . B),(h . C),(h . D)} by A5, XBOOLE_0:def_10; ::_thesis: verum end; theorem :: BVFUNC14:18 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D being a_partition of Y for z, u being Element of Y for h being Function st G is independent & G = {A,B,C,D} & A <> B & A <> C & A <> D & B <> C & B <> D & C <> D holds EqClass (u,((B '/\' C) '/\' D)) meets EqClass (z,A) proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C, D being a_partition of Y for z, u being Element of Y for h being Function st G is independent & G = {A,B,C,D} & A <> B & A <> C & A <> D & B <> C & B <> D & C <> D holds EqClass (u,((B '/\' C) '/\' D)) meets EqClass (z,A) let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C, D being a_partition of Y for z, u being Element of Y for h being Function st G is independent & G = {A,B,C,D} & A <> B & A <> C & A <> D & B <> C & B <> D & C <> D holds EqClass (u,((B '/\' C) '/\' D)) meets EqClass (z,A) let A, B, C, D be a_partition of Y; ::_thesis: for z, u being Element of Y for h being Function st G is independent & G = {A,B,C,D} & A <> B & A <> C & A <> D & B <> C & B <> D & C <> D holds EqClass (u,((B '/\' C) '/\' D)) meets EqClass (z,A) let z, u be Element of Y; ::_thesis: for h being Function st G is independent & G = {A,B,C,D} & A <> B & A <> C & A <> D & B <> C & B <> D & C <> D holds EqClass (u,((B '/\' C) '/\' D)) meets EqClass (z,A) let h be Function; ::_thesis: ( G is independent & G = {A,B,C,D} & A <> B & A <> C & A <> D & B <> C & B <> D & C <> D implies EqClass (u,((B '/\' C) '/\' D)) meets EqClass (z,A) ) assume that A1: G is independent and A2: G = {A,B,C,D} and A3: ( A <> B & A <> C & A <> D & B <> C & B <> D & C <> D ) ; ::_thesis: EqClass (u,((B '/\' C) '/\' D)) meets EqClass (z,A) set h = (((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A))); A4: ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . B = EqClass (u,B) by A3, Th15; A5: ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . D = EqClass (u,D) by A3, Th15; A6: ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . C = EqClass (u,C) by A3, Th15; A7: rng ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) = {(((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . A),(((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . B),(((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . C),(((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . D)} by A2, Th17; rng ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) c= bool Y proof let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in rng ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) or t in bool Y ) assume A8: t in rng ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) ; ::_thesis: t in bool Y percases ( t = ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . A or t = ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . B or t = ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . C or t = ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . D ) by A7, A8, ENUMSET1:def_2; suppose t = ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . A ; ::_thesis: t in bool Y then t = EqClass (z,A) by FUNCT_7:94; hence t in bool Y ; ::_thesis: verum end; suppose t = ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . B ; ::_thesis: t in bool Y hence t in bool Y by A4; ::_thesis: verum end; suppose t = ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . C ; ::_thesis: t in bool Y hence t in bool Y by A6; ::_thesis: verum end; suppose t = ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . D ; ::_thesis: t in bool Y hence t in bool Y by A5; ::_thesis: verum end; end; end; then reconsider FF = rng ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) as Subset-Family of Y ; A9: dom ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) = G by A2, Th16; for d being set st d in G holds ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . d in d proof let d be set ; ::_thesis: ( d in G implies ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . d in d ) assume A10: d in G ; ::_thesis: ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . d in d percases ( d = A or d = B or d = C or d = D ) by A2, A10, ENUMSET1:def_2; supposeA11: d = A ; ::_thesis: ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . d in d ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . A = EqClass (z,A) by FUNCT_7:94; hence ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . d in d by A11; ::_thesis: verum end; supposeA12: d = B ; ::_thesis: ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . d in d ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . B = EqClass (u,B) by A3, Th15; hence ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . d in d by A12; ::_thesis: verum end; supposeA13: d = C ; ::_thesis: ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . d in d ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . C = EqClass (u,C) by A3, Th15; hence ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . d in d by A13; ::_thesis: verum end; supposeA14: d = D ; ::_thesis: ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . d in d ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . D = EqClass (u,D) by A3, Th15; hence ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . d in d by A14; ::_thesis: verum end; end; end; then Intersect FF <> {} by A1, A9, BVFUNC_2:def_5; then consider m being set such that A15: m in Intersect FF by XBOOLE_0:def_1; A in dom ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) by A2, A9, ENUMSET1:def_2; then A16: ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . A in rng ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then A17: m in meet FF by A15, SETFAM_1:def_9; D in dom ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) by A2, A9, ENUMSET1:def_2; then ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . D in rng ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then A18: m in ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . D by A17, SETFAM_1:def_1; C in dom ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) by A2, A9, ENUMSET1:def_2; then ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . C in rng ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then A19: m in ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . C by A17, SETFAM_1:def_1; B in dom ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) by A2, A9, ENUMSET1:def_2; then ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . B in rng ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then m in ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . B by A17, SETFAM_1:def_1; then m in (EqClass (u,B)) /\ (EqClass (u,C)) by A4, A6, A19, XBOOLE_0:def_4; then A20: m in ((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D)) by A5, A18, XBOOLE_0:def_4; ( ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . A = EqClass (z,A) & m in ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . A ) by A16, A17, FUNCT_7:94, SETFAM_1:def_1; then m in (((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (z,A)) by A20, XBOOLE_0:def_4; then A21: ((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D)) meets EqClass (z,A) by XBOOLE_0:4; EqClass (u,((B '/\' C) '/\' D)) = (EqClass (u,(B '/\' C))) /\ (EqClass (u,D)) by Th1; hence EqClass (u,((B '/\' C) '/\' D)) meets EqClass (z,A) by A21, Th1; ::_thesis: verum end; theorem :: BVFUNC14:19 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D being a_partition of Y for z, u being Element of Y st G is independent & G = {A,B,C,D} & A <> B & A <> C & A <> D & B <> C & B <> D & C <> D & EqClass (z,(C '/\' D)) = EqClass (u,(C '/\' D)) holds EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G))) proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C, D being a_partition of Y for z, u being Element of Y st G is independent & G = {A,B,C,D} & A <> B & A <> C & A <> D & B <> C & B <> D & C <> D & EqClass (z,(C '/\' D)) = EqClass (u,(C '/\' D)) holds EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G))) let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C, D being a_partition of Y for z, u being Element of Y st G is independent & G = {A,B,C,D} & A <> B & A <> C & A <> D & B <> C & B <> D & C <> D & EqClass (z,(C '/\' D)) = EqClass (u,(C '/\' D)) holds EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G))) let A, B, C, D be a_partition of Y; ::_thesis: for z, u being Element of Y st G is independent & G = {A,B,C,D} & A <> B & A <> C & A <> D & B <> C & B <> D & C <> D & EqClass (z,(C '/\' D)) = EqClass (u,(C '/\' D)) holds EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G))) let z, u be Element of Y; ::_thesis: ( G is independent & G = {A,B,C,D} & A <> B & A <> C & A <> D & B <> C & B <> D & C <> D & EqClass (z,(C '/\' D)) = EqClass (u,(C '/\' D)) implies EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G))) ) assume that A1: G is independent and A2: G = {A,B,C,D} and A3: A <> B and A4: ( A <> C & A <> D ) and A5: ( B <> C & B <> D ) and A6: C <> D and A7: EqClass (z,(C '/\' D)) = EqClass (u,(C '/\' D)) ; ::_thesis: EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G))) set h = (((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A))); set H = EqClass (z,(CompF (B,G))); A8: A '/\' (C '/\' D) = (A '/\' C) '/\' D by PARTIT1:14; A9: rng ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) = {(((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . A),(((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . B),(((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . C),(((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . D)} by A2, Th17; rng ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) c= bool Y proof let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in rng ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) or t in bool Y ) assume A10: t in rng ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) ; ::_thesis: t in bool Y percases ( t = ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . A or t = ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . B or t = ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . C or t = ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . D ) by A9, A10, ENUMSET1:def_2; suppose t = ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . A ; ::_thesis: t in bool Y then t = EqClass (z,A) by FUNCT_7:94; hence t in bool Y ; ::_thesis: verum end; suppose t = ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . B ; ::_thesis: t in bool Y then t = EqClass (u,B) by A3, A4, A5, A6, Th15; hence t in bool Y ; ::_thesis: verum end; suppose t = ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . C ; ::_thesis: t in bool Y then t = EqClass (u,C) by A3, A4, A5, A6, Th15; hence t in bool Y ; ::_thesis: verum end; suppose t = ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . D ; ::_thesis: t in bool Y then t = EqClass (u,D) by A3, A4, A5, A6, Th15; hence t in bool Y ; ::_thesis: verum end; end; end; then reconsider FF = rng ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) as Subset-Family of Y ; set I = EqClass (z,A); set GG = EqClass (u,((B '/\' C) '/\' D)); A11: EqClass (u,((B '/\' C) '/\' D)) = (EqClass (u,(B '/\' C))) /\ (EqClass (u,D)) by Th1; A12: for d being set st d in G holds ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . d in d proof let d be set ; ::_thesis: ( d in G implies ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . d in d ) assume A13: d in G ; ::_thesis: ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . d in d percases ( d = A or d = B or d = C or d = D ) by A2, A13, ENUMSET1:def_2; supposeA14: d = A ; ::_thesis: ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . d in d ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . A = EqClass (z,A) by FUNCT_7:94; hence ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . d in d by A14; ::_thesis: verum end; supposeA15: d = B ; ::_thesis: ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . d in d ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . B = EqClass (u,B) by A3, A4, A5, A6, Th15; hence ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . d in d by A15; ::_thesis: verum end; supposeA16: d = C ; ::_thesis: ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . d in d ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . C = EqClass (u,C) by A3, A4, A5, A6, Th15; hence ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . d in d by A16; ::_thesis: verum end; supposeA17: d = D ; ::_thesis: ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . d in d ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . D = EqClass (u,D) by A3, A4, A5, A6, Th15; hence ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . d in d by A17; ::_thesis: verum end; end; end; dom ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) = G by A2, Th16; then Intersect FF <> {} by A1, A12, BVFUNC_2:def_5; then consider m being set such that A18: m in Intersect FF by XBOOLE_0:def_1; A19: dom ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) = G by A2, Th16; then A in dom ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) by A2, ENUMSET1:def_2; then A20: ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . A in rng ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then A21: m in meet FF by A18, SETFAM_1:def_9; then A22: ( ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . A = EqClass (z,A) & m in ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . A ) by A20, FUNCT_7:94, SETFAM_1:def_1; D in dom ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) by A2, A19, ENUMSET1:def_2; then ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . D in rng ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then A23: m in ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . D by A21, SETFAM_1:def_1; C in dom ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) by A2, A19, ENUMSET1:def_2; then ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . C in rng ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then A24: m in ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . C by A21, SETFAM_1:def_1; B in dom ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) by A2, A19, ENUMSET1:def_2; then ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . B in rng ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then A25: m in ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . B by A21, SETFAM_1:def_1; ( ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . B = EqClass (u,B) & ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . C = EqClass (u,C) ) by A3, A4, A5, A6, Th15; then A26: m in (EqClass (u,B)) /\ (EqClass (u,C)) by A25, A24, XBOOLE_0:def_4; ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (A .--> (EqClass (z,A)))) . D = EqClass (u,D) by A3, A4, A5, A6, Th15; then m in ((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D)) by A23, A26, XBOOLE_0:def_4; then m in (((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (z,A)) by A22, XBOOLE_0:def_4; then (EqClass (u,((B '/\' C) '/\' D))) /\ (EqClass (z,A)) <> {} by A11, Th1; then consider p being set such that A27: p in (EqClass (u,((B '/\' C) '/\' D))) /\ (EqClass (z,A)) by XBOOLE_0:def_1; reconsider p = p as Element of Y by A27; set K = EqClass (p,(C '/\' D)); A28: p in EqClass (u,((B '/\' C) '/\' D)) by A27, XBOOLE_0:def_4; set L = EqClass (z,(C '/\' D)); A29: z in EqClass (z,A) by EQREL_1:def_6; EqClass (u,((B '/\' C) '/\' D)) = EqClass (u,(B '/\' (C '/\' D))) by PARTIT1:14; then A30: EqClass (u,((B '/\' C) '/\' D)) c= EqClass (u,(C '/\' D)) by BVFUNC11:3; p in EqClass (p,(C '/\' D)) by EQREL_1:def_6; then EqClass (p,(C '/\' D)) meets EqClass (z,(C '/\' D)) by A7, A30, A28, XBOOLE_0:3; then EqClass (p,(C '/\' D)) = EqClass (z,(C '/\' D)) by EQREL_1:41; then z in EqClass (p,(C '/\' D)) by EQREL_1:def_6; then A31: z in (EqClass (z,A)) /\ (EqClass (p,(C '/\' D))) by A29, XBOOLE_0:def_4; A32: ( p in EqClass (p,(C '/\' D)) & p in EqClass (z,A) ) by A27, EQREL_1:def_6, XBOOLE_0:def_4; then p in (EqClass (z,A)) /\ (EqClass (p,(C '/\' D))) by XBOOLE_0:def_4; then ( (EqClass (z,A)) /\ (EqClass (p,(C '/\' D))) in INTERSECTION (A,(C '/\' D)) & not (EqClass (z,A)) /\ (EqClass (p,(C '/\' D))) in {{}} ) by SETFAM_1:def_5, TARSKI:def_1; then A33: (EqClass (z,A)) /\ (EqClass (p,(C '/\' D))) in (INTERSECTION (A,(C '/\' D))) \ {{}} by XBOOLE_0:def_5; CompF (B,G) = (A '/\' C) '/\' D by A2, A3, A5, Th8; then (EqClass (z,A)) /\ (EqClass (p,(C '/\' D))) in CompF (B,G) by A33, A8, PARTIT1:def_4; then A34: ( (EqClass (z,A)) /\ (EqClass (p,(C '/\' D))) = EqClass (z,(CompF (B,G))) or (EqClass (z,A)) /\ (EqClass (p,(C '/\' D))) misses EqClass (z,(CompF (B,G))) ) by EQREL_1:def_4; z in EqClass (z,(CompF (B,G))) by EQREL_1:def_6; then p in EqClass (z,(CompF (B,G))) by A32, A31, A34, XBOOLE_0:3, XBOOLE_0:def_4; then p in (EqClass (u,((B '/\' C) '/\' D))) /\ (EqClass (z,(CompF (B,G)))) by A28, XBOOLE_0:def_4; then EqClass (u,((B '/\' C) '/\' D)) meets EqClass (z,(CompF (B,G))) by XBOOLE_0:4; hence EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G))) by A2, A3, A4, Th7; ::_thesis: verum end; theorem :: BVFUNC14:20 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C being a_partition of Y for z, u being Element of Y st G is independent & G = {A,B,C} & A <> B & B <> C & C <> A & EqClass (z,C) = EqClass (u,C) holds EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G))) proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C being a_partition of Y for z, u being Element of Y st G is independent & G = {A,B,C} & A <> B & B <> C & C <> A & EqClass (z,C) = EqClass (u,C) holds EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G))) let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C being a_partition of Y for z, u being Element of Y st G is independent & G = {A,B,C} & A <> B & B <> C & C <> A & EqClass (z,C) = EqClass (u,C) holds EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G))) let A, B, C be a_partition of Y; ::_thesis: for z, u being Element of Y st G is independent & G = {A,B,C} & A <> B & B <> C & C <> A & EqClass (z,C) = EqClass (u,C) holds EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G))) let z, u be Element of Y; ::_thesis: ( G is independent & G = {A,B,C} & A <> B & B <> C & C <> A & EqClass (z,C) = EqClass (u,C) implies EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G))) ) assume that A1: G is independent and A2: G = {A,B,C} and A3: A <> B and A4: B <> C and A5: C <> A and A6: EqClass (z,C) = EqClass (u,C) ; ::_thesis: EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G))) set h = ((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A))); A7: dom (A .--> (EqClass (z,A))) = {A} by FUNCOP_1:13; then A in dom (A .--> (EqClass (z,A))) by TARSKI:def_1; then (((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . A = (A .--> (EqClass (z,A))) . A by FUNCT_4:13; then A8: (((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . A = EqClass (z,A) by FUNCOP_1:72; set H = EqClass (z,(CompF (B,G))); set I = EqClass (z,A); set GG = EqClass (u,(B '/\' C)); A9: (EqClass (u,(B '/\' C))) /\ (EqClass (z,A)) = ((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (z,A)) by Th1; dom ((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) = (dom (B .--> (EqClass (u,B)))) \/ (dom (C .--> (EqClass (u,C)))) by FUNCT_4:def_1; then A10: dom (((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) = ((dom (B .--> (EqClass (u,B)))) \/ (dom (C .--> (EqClass (u,C))))) \/ (dom (A .--> (EqClass (z,A)))) by FUNCT_4:def_1; A11: dom (C .--> (EqClass (u,C))) = {C} by FUNCOP_1:13; then A12: C in dom (C .--> (EqClass (u,C))) by TARSKI:def_1; not B in dom (A .--> (EqClass (z,A))) by A3, A7, TARSKI:def_1; then A13: (((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . B = ((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) . B by FUNCT_4:11; not B in dom (C .--> (EqClass (u,C))) by A4, A11, TARSKI:def_1; then (((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . B = (B .--> (EqClass (u,B))) . B by A13, FUNCT_4:11; then A14: (((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . B = EqClass (u,B) by FUNCOP_1:72; not C in dom (A .--> (EqClass (z,A))) by A5, A7, TARSKI:def_1; then (((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . C = ((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) . C by FUNCT_4:11; then (((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . C = (C .--> (EqClass (u,C))) . C by A12, FUNCT_4:13; then A15: (((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . C = EqClass (u,C) by FUNCOP_1:72; dom (B .--> (EqClass (u,B))) = {B} by FUNCOP_1:13; then A16: dom (((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) = ({A} \/ {B}) \/ {C} by A10, A11, A7, XBOOLE_1:4 .= {A,B} \/ {C} by ENUMSET1:1 .= {A,B,C} by ENUMSET1:3 ; A17: rng (((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) c= {((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . A),((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . B),((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . C)} proof let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in rng (((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) or t in {((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . A),((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . B),((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . C)} ) assume t in rng (((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) ; ::_thesis: t in {((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . A),((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . B),((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . C)} then consider x1 being set such that A18: x1 in dom (((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) and A19: t = (((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . x1 by FUNCT_1:def_3; now__::_thesis:_(_(_x1_=_A_&_t_in_{((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(A_.-->_(EqClass_(z,A))))_._A),((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(A_.-->_(EqClass_(z,A))))_._B),((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(A_.-->_(EqClass_(z,A))))_._C)}_)_or_(_x1_=_B_&_t_in_{((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(A_.-->_(EqClass_(z,A))))_._A),((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(A_.-->_(EqClass_(z,A))))_._B),((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(A_.-->_(EqClass_(z,A))))_._C)}_)_or_(_x1_=_C_&_t_in_{((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(A_.-->_(EqClass_(z,A))))_._A),((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(A_.-->_(EqClass_(z,A))))_._B),((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(A_.-->_(EqClass_(z,A))))_._C)}_)_) percases ( x1 = A or x1 = B or x1 = C ) by A16, A18, ENUMSET1:def_1; case x1 = A ; ::_thesis: t in {((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . A),((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . B),((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . C)} hence t in {((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . A),((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . B),((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . C)} by A19, ENUMSET1:def_1; ::_thesis: verum end; case x1 = B ; ::_thesis: t in {((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . A),((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . B),((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . C)} hence t in {((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . A),((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . B),((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . C)} by A19, ENUMSET1:def_1; ::_thesis: verum end; case x1 = C ; ::_thesis: t in {((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . A),((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . B),((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . C)} hence t in {((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . A),((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . B),((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . C)} by A19, ENUMSET1:def_1; ::_thesis: verum end; end; end; hence t in {((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . A),((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . B),((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . C)} ; ::_thesis: verum end; rng (((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) c= bool Y proof let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in rng (((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) or t in bool Y ) assume A20: t in rng (((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) ; ::_thesis: t in bool Y now__::_thesis:_(_(_t_=_(((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(A_.-->_(EqClass_(z,A))))_._A_&_t_in_bool_Y_)_or_(_t_=_(((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(A_.-->_(EqClass_(z,A))))_._B_&_t_in_bool_Y_)_or_(_t_=_(((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(A_.-->_(EqClass_(z,A))))_._C_&_t_in_bool_Y_)_) percases ( t = (((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . A or t = (((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . B or t = (((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . C ) by A17, A20, ENUMSET1:def_1; case t = (((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . A ; ::_thesis: t in bool Y hence t in bool Y by A8; ::_thesis: verum end; case t = (((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . B ; ::_thesis: t in bool Y hence t in bool Y by A14; ::_thesis: verum end; case t = (((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . C ; ::_thesis: t in bool Y hence t in bool Y by A15; ::_thesis: verum end; end; end; hence t in bool Y ; ::_thesis: verum end; then reconsider FF = rng (((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) as Subset-Family of Y ; A21: z in EqClass (z,(CompF (B,G))) by EQREL_1:def_6; for d being set st d in G holds (((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . d in d proof let d be set ; ::_thesis: ( d in G implies (((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . d in d ) assume A22: d in G ; ::_thesis: (((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . d in d now__::_thesis:_(_(_d_=_A_&_(((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(A_.-->_(EqClass_(z,A))))_._d_in_d_)_or_(_d_=_B_&_(((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(A_.-->_(EqClass_(z,A))))_._d_in_d_)_or_(_d_=_C_&_(((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(A_.-->_(EqClass_(z,A))))_._d_in_d_)_) percases ( d = A or d = B or d = C ) by A2, A22, ENUMSET1:def_1; case d = A ; ::_thesis: (((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . d in d hence (((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . d in d by A8; ::_thesis: verum end; case d = B ; ::_thesis: (((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . d in d hence (((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . d in d by A14; ::_thesis: verum end; case d = C ; ::_thesis: (((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . d in d hence (((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . d in d by A15; ::_thesis: verum end; end; end; hence (((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . d in d ; ::_thesis: verum end; then Intersect FF <> {} by A1, A2, A16, BVFUNC_2:def_5; then consider m being set such that A23: m in Intersect FF by XBOOLE_0:def_1; A in dom (((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) by A16, ENUMSET1:def_1; then A24: (((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . A in rng (((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then A25: Intersect FF = meet (rng (((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A))))) by SETFAM_1:def_9; C in dom (((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) by A16, ENUMSET1:def_1; then (((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . C in rng (((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then A26: m in (((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . C by A25, A23, SETFAM_1:def_1; B in dom (((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) by A16, ENUMSET1:def_1; then (((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . B in rng (((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then m in (((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . B by A25, A23, SETFAM_1:def_1; then A27: m in (EqClass (u,B)) /\ (EqClass (u,C)) by A14, A15, A26, XBOOLE_0:def_4; m in (((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (A .--> (EqClass (z,A)))) . A by A24, A25, A23, SETFAM_1:def_1; then (EqClass (u,(B '/\' C))) /\ (EqClass (z,A)) <> {} by A8, A9, A27, XBOOLE_0:def_4; then consider p being set such that A28: p in (EqClass (u,(B '/\' C))) /\ (EqClass (z,A)) by XBOOLE_0:def_1; reconsider p = p as Element of Y by A28; set K = EqClass (p,C); A29: (EqClass (z,A)) /\ (EqClass (p,C)) in INTERSECTION (A,C) by SETFAM_1:def_5; set L = EqClass (z,C); A30: p in EqClass (p,C) by EQREL_1:def_6; A31: p in EqClass (u,(B '/\' C)) by A28, XBOOLE_0:def_4; ( p in EqClass (p,C) & p in EqClass (z,A) ) by A28, EQREL_1:def_6, XBOOLE_0:def_4; then A32: p in (EqClass (z,A)) /\ (EqClass (p,C)) by XBOOLE_0:def_4; then not (EqClass (z,A)) /\ (EqClass (p,C)) in {{}} by TARSKI:def_1; then (EqClass (z,A)) /\ (EqClass (p,C)) in (INTERSECTION (A,C)) \ {{}} by A29, XBOOLE_0:def_5; then A33: (EqClass (z,A)) /\ (EqClass (p,C)) in A '/\' C by PARTIT1:def_4; EqClass (u,(B '/\' C)) c= EqClass (z,C) by A6, BVFUNC11:3; then EqClass (p,C) meets EqClass (z,C) by A31, A30, XBOOLE_0:3; then EqClass (p,C) = EqClass (z,C) by EQREL_1:41; then A34: z in EqClass (p,C) by EQREL_1:def_6; z in EqClass (z,A) by EQREL_1:def_6; then A35: z in (EqClass (z,A)) /\ (EqClass (p,C)) by A34, XBOOLE_0:def_4; CompF (B,G) = A '/\' C by A2, A3, A4, Th5; then A36: ( (EqClass (z,A)) /\ (EqClass (p,C)) = EqClass (z,(CompF (B,G))) or (EqClass (z,A)) /\ (EqClass (p,C)) misses EqClass (z,(CompF (B,G))) ) by A33, EQREL_1:def_4; EqClass (u,(B '/\' C)) = EqClass (u,(CompF (A,G))) by A2, A3, A5, Th4; hence EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G))) by A32, A31, A35, A21, A36, XBOOLE_0:3; ::_thesis: verum end; theorem Th21: :: BVFUNC14:21 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E being a_partition of Y st G = {A,B,C,D,E} & A <> B & A <> C & A <> D & A <> E holds CompF (A,G) = ((B '/\' C) '/\' D) '/\' E proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C, D, E being a_partition of Y st G = {A,B,C,D,E} & A <> B & A <> C & A <> D & A <> E holds CompF (A,G) = ((B '/\' C) '/\' D) '/\' E let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C, D, E being a_partition of Y st G = {A,B,C,D,E} & A <> B & A <> C & A <> D & A <> E holds CompF (A,G) = ((B '/\' C) '/\' D) '/\' E let A, B, C, D, E be a_partition of Y; ::_thesis: ( G = {A,B,C,D,E} & A <> B & A <> C & A <> D & A <> E implies CompF (A,G) = ((B '/\' C) '/\' D) '/\' E ) assume that A1: G = {A,B,C,D,E} and A2: A <> B and A3: A <> C and A4: A <> D and A5: A <> E ; ::_thesis: CompF (A,G) = ((B '/\' C) '/\' D) '/\' E percases ( B = C or B = D or B = E or C = D or C = E or D = E or ( B <> C & B <> D & B <> E & C <> D & C <> E & D <> E ) ) ; supposeA6: B = C ; ::_thesis: CompF (A,G) = ((B '/\' C) '/\' D) '/\' E then G = {A,B,B,D} \/ {E} by A1, ENUMSET1:10 .= {B,B,A,D} \/ {E} by ENUMSET1:67 .= {B,B,A,D,E} by ENUMSET1:10 .= {B,A,D,E} by ENUMSET1:32 .= {A,B,D,E} by ENUMSET1:65 ; hence CompF (A,G) = (B '/\' D) '/\' E by A2, A4, A5, Th7 .= ((B '/\' C) '/\' D) '/\' E by A6, PARTIT1:13 ; ::_thesis: verum end; supposeA7: B = D ; ::_thesis: CompF (A,G) = ((B '/\' C) '/\' D) '/\' E then G = {A,B,C,B} \/ {E} by A1, ENUMSET1:10 .= {B,B,A,C} \/ {E} by ENUMSET1:69 .= {B,B,A,C,E} by ENUMSET1:10 .= {B,A,C,E} by ENUMSET1:32 .= {A,B,C,E} by ENUMSET1:65 ; hence CompF (A,G) = (B '/\' C) '/\' E by A2, A3, A5, Th7 .= ((B '/\' D) '/\' C) '/\' E by A7, PARTIT1:13 .= ((B '/\' C) '/\' D) '/\' E by PARTIT1:14 ; ::_thesis: verum end; supposeA8: B = E ; ::_thesis: CompF (A,G) = ((B '/\' C) '/\' D) '/\' E then G = {A} \/ {B,C,D,B} by A1, ENUMSET1:7 .= {A} \/ {B,B,C,D} by ENUMSET1:63 .= {A} \/ {B,C,D} by ENUMSET1:31 .= {A,B,C,D} by ENUMSET1:4 ; hence CompF (A,G) = (B '/\' C) '/\' D by A2, A3, A4, Th7 .= ((B '/\' E) '/\' C) '/\' D by A8, PARTIT1:13 .= (B '/\' E) '/\' (C '/\' D) by PARTIT1:14 .= (B '/\' (C '/\' D)) '/\' E by PARTIT1:14 .= ((B '/\' C) '/\' D) '/\' E by PARTIT1:14 ; ::_thesis: verum end; supposeA9: C = D ; ::_thesis: CompF (A,G) = ((B '/\' C) '/\' D) '/\' E then G = {A,B,C,C} \/ {E} by A1, ENUMSET1:10 .= {C,C,A,B} \/ {E} by ENUMSET1:73 .= {C,A,B} \/ {E} by ENUMSET1:31 .= {C,A,B,E} by ENUMSET1:6 .= {A,B,C,E} by ENUMSET1:67 ; hence CompF (A,G) = (B '/\' C) '/\' E by A2, A3, A5, Th7 .= (B '/\' (C '/\' D)) '/\' E by A9, PARTIT1:13 .= ((B '/\' C) '/\' D) '/\' E by PARTIT1:14 ; ::_thesis: verum end; supposeA10: C = E ; ::_thesis: CompF (A,G) = ((B '/\' C) '/\' D) '/\' E then G = {A} \/ {B,C,D,C} by A1, ENUMSET1:7 .= {A} \/ {C,C,B,D} by ENUMSET1:72 .= {A} \/ {C,B,D} by ENUMSET1:31 .= {A,C,B,D} by ENUMSET1:4 .= {A,B,C,D} by ENUMSET1:62 ; hence CompF (A,G) = (B '/\' C) '/\' D by A2, A3, A4, Th7 .= (B '/\' (C '/\' E)) '/\' D by A10, PARTIT1:13 .= B '/\' ((C '/\' E) '/\' D) by PARTIT1:14 .= B '/\' ((C '/\' D) '/\' E) by PARTIT1:14 .= (B '/\' (C '/\' D)) '/\' E by PARTIT1:14 .= ((B '/\' C) '/\' D) '/\' E by PARTIT1:14 ; ::_thesis: verum end; supposeA11: D = E ; ::_thesis: CompF (A,G) = ((B '/\' C) '/\' D) '/\' E then G = {A} \/ {B,C,D,D} by A1, ENUMSET1:7 .= {A} \/ {D,D,B,C} by ENUMSET1:73 .= {A} \/ {D,B,C} by ENUMSET1:31 .= {A,D,B,C} by ENUMSET1:4 .= {A,B,C,D} by ENUMSET1:63 ; hence CompF (A,G) = (B '/\' C) '/\' D by A2, A3, A4, Th7 .= (B '/\' C) '/\' (D '/\' E) by A11, PARTIT1:13 .= B '/\' (C '/\' (D '/\' E)) by PARTIT1:14 .= B '/\' ((C '/\' D) '/\' E) by PARTIT1:14 .= (B '/\' (C '/\' D)) '/\' E by PARTIT1:14 .= ((B '/\' C) '/\' D) '/\' E by PARTIT1:14 ; ::_thesis: verum end; supposeA12: ( B <> C & B <> D & B <> E & C <> D & C <> E & D <> E ) ; ::_thesis: CompF (A,G) = ((B '/\' C) '/\' D) '/\' E A13: ( not D in {A} & not E in {A} ) by A4, A5, TARSKI:def_1; A14: not B in {A} by A2, TARSKI:def_1; G \ {A} = ({A} \/ {B,C,D,E}) \ {A} by A1, ENUMSET1:7; then A15: G \ {A} = ({A} \ {A}) \/ ({B,C,D,E} \ {A}) by XBOOLE_1:42; A16: not C in {A} by A3, TARSKI:def_1; A in {A} by TARSKI:def_1; then A17: {A} \ {A} = {} by ZFMISC_1:60; {B,C,D,E} \ {A} = ({B} \/ {C,D,E}) \ {A} by ENUMSET1:4 .= ({B} \ {A}) \/ ({C,D,E} \ {A}) by XBOOLE_1:42 .= {B} \/ ({C,D,E} \ {A}) by A14, ZFMISC_1:59 .= {B} \/ (({C} \/ {D,E}) \ {A}) by ENUMSET1:2 .= {B} \/ (({C} \ {A}) \/ ({D,E} \ {A})) by XBOOLE_1:42 .= {B} \/ (({C} \ {A}) \/ {D,E}) by A13, ZFMISC_1:63 .= {B} \/ ({C} \/ {D,E}) by A16, ZFMISC_1:59 .= {B} \/ {C,D,E} by ENUMSET1:2 ; then A18: G \ {A} = ({A} \ {A}) \/ {B,C,D,E} by A15, ENUMSET1:4; A19: ((B '/\' C) '/\' D) '/\' E c= '/\' (G \ {A}) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in ((B '/\' C) '/\' D) '/\' E or x in '/\' (G \ {A}) ) assume A20: x in ((B '/\' C) '/\' D) '/\' E ; ::_thesis: x in '/\' (G \ {A}) then A21: x <> {} by EQREL_1:def_4; x in (INTERSECTION (((B '/\' C) '/\' D),E)) \ {{}} by A20, PARTIT1:def_4; then consider bcd, e being set such that A22: bcd in (B '/\' C) '/\' D and A23: e in E and A24: x = bcd /\ e by SETFAM_1:def_5; bcd in (INTERSECTION ((B '/\' C),D)) \ {{}} by A22, PARTIT1:def_4; then consider bc, d being set such that A25: bc in B '/\' C and A26: d in D and A27: bcd = bc /\ d by SETFAM_1:def_5; bc in (INTERSECTION (B,C)) \ {{}} by A25, PARTIT1:def_4; then consider b, c being set such that A28: b in B and A29: c in C and A30: bc = b /\ c by SETFAM_1:def_5; set h = (((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e); A31: dom (C .--> c) = {C} by FUNCOP_1:13; then A32: C in dom (C .--> c) by TARSKI:def_1; A33: dom (D .--> d) = {D} by FUNCOP_1:13; then A34: D in dom (D .--> d) by TARSKI:def_1; A35: not C in dom (D .--> d) by A12, A33, TARSKI:def_1; A36: dom (E .--> e) = {E} by FUNCOP_1:13; then E in dom (E .--> e) by TARSKI:def_1; then A37: ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . E = (E .--> e) . E by FUNCT_4:13; then A38: ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . E = e by FUNCOP_1:72; not C in dom (E .--> e) by A12, A36, TARSKI:def_1; then ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . C = (((B .--> b) +* (C .--> c)) +* (D .--> d)) . C by FUNCT_4:11; then ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . C = ((B .--> b) +* (C .--> c)) . C by A35, FUNCT_4:11; then A39: ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . C = (C .--> c) . C by A32, FUNCT_4:13; then A40: ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . C = c by FUNCOP_1:72; not D in dom (E .--> e) by A12, A36, TARSKI:def_1; then ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . D = (((B .--> b) +* (C .--> c)) +* (D .--> d)) . D by FUNCT_4:11; then A41: ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . D = (D .--> d) . D by A34, FUNCT_4:13; then A42: ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . D = d by FUNCOP_1:72; A43: not B in dom (C .--> c) by A12, A31, TARSKI:def_1; A44: not B in dom (D .--> d) by A12, A33, TARSKI:def_1; not B in dom (E .--> e) by A12, A36, TARSKI:def_1; then ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . B = (((B .--> b) +* (C .--> c)) +* (D .--> d)) . B by FUNCT_4:11; then ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . B = ((B .--> b) +* (C .--> c)) . B by A44, FUNCT_4:11; then A45: ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . B = (B .--> b) . B by A43, FUNCT_4:11; then A46: ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . B = b by FUNCOP_1:72; A47: for p being set st p in G \ {A} holds ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . p in p proof let p be set ; ::_thesis: ( p in G \ {A} implies ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . p in p ) assume A48: p in G \ {A} ; ::_thesis: ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . p in p now__::_thesis:_(_(_p_=_D_&_((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_._p_in_p_)_or_(_p_=_B_&_((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_._p_in_p_)_or_(_p_=_C_&_((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_._p_in_p_)_or_(_p_=_E_&_((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_._p_in_p_)_) percases ( p = D or p = B or p = C or p = E ) by A15, A17, A48, ENUMSET1:def_2; case p = D ; ::_thesis: ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . p in p hence ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . p in p by A26, A41, FUNCOP_1:72; ::_thesis: verum end; case p = B ; ::_thesis: ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . p in p hence ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . p in p by A28, A45, FUNCOP_1:72; ::_thesis: verum end; case p = C ; ::_thesis: ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . p in p hence ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . p in p by A29, A39, FUNCOP_1:72; ::_thesis: verum end; case p = E ; ::_thesis: ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . p in p hence ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . p in p by A23, A37, FUNCOP_1:72; ::_thesis: verum end; end; end; hence ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . p in p ; ::_thesis: verum end; 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 A49: 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; dom (B .--> b) = {B} by FUNCOP_1:13; then A50: dom ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) = (({B} \/ {C}) \/ {D}) \/ {E} by A49, A31, A33, FUNCOP_1:13 .= ({B,C} \/ {D}) \/ {E} by ENUMSET1:1 .= {B,C,D} \/ {E} by ENUMSET1:3 .= {B,C,D,E} by ENUMSET1:6 ; then A51: D in dom ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) by ENUMSET1:def_2; A52: rng ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) c= {(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . D),(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . B),(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . C),(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . E)} proof let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in rng ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) or t in {(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . D),(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . B),(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . C),(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . E)} ) assume t in rng ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) ; ::_thesis: t in {(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . D),(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . B),(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . C),(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . E)} then consider x1 being set such that A53: x1 in dom ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) and A54: t = ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . x1 by FUNCT_1:def_3; now__::_thesis:_(_(_x1_=_D_&_t_in_{(((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_._D),(((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_._B),(((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_._C),(((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_._E)}_)_or_(_x1_=_B_&_t_in_{(((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_._D),(((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_._B),(((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_._C),(((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_._E)}_)_or_(_x1_=_C_&_t_in_{(((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_._D),(((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_._B),(((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_._C),(((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_._E)}_)_or_(_x1_=_E_&_t_in_{(((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_._D),(((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_._B),(((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_._C),(((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_._E)}_)_) percases ( x1 = D or x1 = B or x1 = C or x1 = E ) by A50, A53, ENUMSET1:def_2; case x1 = D ; ::_thesis: t in {(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . D),(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . B),(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . C),(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . E)} hence t in {(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . D),(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . B),(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . C),(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . E)} by A54, ENUMSET1:def_2; ::_thesis: verum end; case x1 = B ; ::_thesis: t in {(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . D),(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . B),(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . C),(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . E)} hence t in {(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . D),(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . B),(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . C),(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . E)} by A54, ENUMSET1:def_2; ::_thesis: verum end; case x1 = C ; ::_thesis: t in {(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . D),(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . B),(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . C),(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . E)} hence t in {(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . D),(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . B),(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . C),(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . E)} by A54, ENUMSET1:def_2; ::_thesis: verum end; case x1 = E ; ::_thesis: t in {(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . D),(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . B),(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . C),(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . E)} hence t in {(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . D),(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . B),(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . C),(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . E)} by A54, ENUMSET1:def_2; ::_thesis: verum end; end; end; hence t in {(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . D),(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . B),(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . C),(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . E)} ; ::_thesis: verum end; rng ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) c= bool Y proof let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in rng ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) or t in bool Y ) assume A55: t in rng ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) ; ::_thesis: t in bool Y now__::_thesis:_(_(_t_=_((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_._D_&_t_in_bool_Y_)_or_(_t_=_((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_._B_&_t_in_bool_Y_)_or_(_t_=_((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_._C_&_t_in_bool_Y_)_or_(_t_=_((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_._E_&_t_in_bool_Y_)_) percases ( t = ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . D or t = ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . B or t = ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . C or t = ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . E ) by A52, A55, ENUMSET1:def_2; case t = ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . D ; ::_thesis: t in bool Y hence t in bool Y by A26, A42; ::_thesis: verum end; case t = ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . B ; ::_thesis: t in bool Y hence t in bool Y by A28, A46; ::_thesis: verum end; case t = ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . C ; ::_thesis: t in bool Y hence t in bool Y by A29, A40; ::_thesis: verum end; case t = ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . E ; ::_thesis: t in bool Y hence t in bool Y by A23, A38; ::_thesis: verum end; end; end; hence t in bool Y ; ::_thesis: verum end; then reconsider F = rng ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) as Subset-Family of Y ; A56: C in dom ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) by A50, ENUMSET1:def_2; A57: E in dom ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) by A50, ENUMSET1:def_2; A58: B in dom ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) by A50, ENUMSET1:def_2; A59: {(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . D),(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . B),(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . C),(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . E)} c= rng ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) proof let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in {(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . D),(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . B),(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . C),(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . E)} or t in rng ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) ) assume A60: t in {(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . D),(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . B),(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . C),(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . E)} ; ::_thesis: t in rng ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) now__::_thesis:_(_(_t_=_((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_._D_&_t_in_rng_((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_)_or_(_t_=_((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_._B_&_t_in_rng_((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_)_or_(_t_=_((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_._C_&_t_in_rng_((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_)_or_(_t_=_((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_._E_&_t_in_rng_((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_)_) percases ( t = ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . D or t = ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . B or t = ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . C or t = ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . E ) by A60, ENUMSET1:def_2; case t = ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . D ; ::_thesis: t in rng ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) hence t in rng ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) by A51, FUNCT_1:def_3; ::_thesis: verum end; case t = ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . B ; ::_thesis: t in rng ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) hence t in rng ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) by A58, FUNCT_1:def_3; ::_thesis: verum end; case t = ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . C ; ::_thesis: t in rng ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) hence t in rng ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) by A56, FUNCT_1:def_3; ::_thesis: verum end; case t = ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . E ; ::_thesis: t in rng ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) hence t in rng ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) by A57, FUNCT_1:def_3; ::_thesis: verum end; end; end; hence t in rng ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) ; ::_thesis: verum end; then A61: {(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . D),(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . B),(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . C),(((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) . E)} = rng ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) by A52, XBOOLE_0:def_10; reconsider h = (((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e) as Function ; A62: x c= Intersect F proof let u be set ; :: according to TARSKI:def_3 ::_thesis: ( not u in x or u in Intersect F ) A63: h . D in {(h . D),(h . B),(h . C),(h . E)} by ENUMSET1:def_2; assume A64: u in x ; ::_thesis: u in Intersect F for y being set st y in F holds u in y proof let y be set ; ::_thesis: ( y in F implies u in y ) assume A65: y in F ; ::_thesis: u in y now__::_thesis:_(_(_y_=_h_._D_&_u_in_y_)_or_(_y_=_h_._B_&_u_in_y_)_or_(_y_=_h_._C_&_u_in_y_)_or_(_y_=_h_._E_&_u_in_y_)_) percases ( y = h . D or y = h . B or y = h . C or y = h . E ) by A52, A65, ENUMSET1:def_2; caseA66: y = h . D ; ::_thesis: u in y u in d /\ ((b /\ c) /\ e) by A24, A27, A30, A64, XBOOLE_1:16; hence u in y by A42, A66, XBOOLE_0:def_4; ::_thesis: verum end; caseA67: y = h . B ; ::_thesis: u in y u in (c /\ (d /\ b)) /\ e by A24, A27, A30, A64, 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 u in y by A46, A67, XBOOLE_0:def_4; ::_thesis: verum end; caseA68: y = h . C ; ::_thesis: u in y u in (c /\ (b /\ d)) /\ e by A24, A27, A30, A64, XBOOLE_1:16; then u in c /\ ((b /\ d) /\ e) by XBOOLE_1:16; hence u in y by A40, A68, XBOOLE_0:def_4; ::_thesis: verum end; case y = h . E ; ::_thesis: u in y hence u in y by A24, A38, A64, XBOOLE_0:def_4; ::_thesis: verum end; end; end; hence u in y ; ::_thesis: verum end; then u in meet F by A59, A63, SETFAM_1:def_1; hence u in Intersect F by A59, A63, SETFAM_1:def_9; ::_thesis: verum end; A69: rng h <> {} by A51, FUNCT_1:3; Intersect F c= x proof let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in Intersect F or t in x ) assume t in Intersect F ; ::_thesis: t in x then A70: t in meet (rng h) by A69, SETFAM_1:def_9; h . D in rng h by A61, ENUMSET1:def_2; then A71: t in h . D by A70, SETFAM_1:def_1; h . C in rng h by A61, ENUMSET1:def_2; then A72: t in h . C by A70, SETFAM_1:def_1; h . B in rng h by A61, ENUMSET1:def_2; then t in h . B by A70, SETFAM_1:def_1; then t in b /\ c by A46, A40, A72, XBOOLE_0:def_4; then A73: t in (b /\ c) /\ d by A42, A71, XBOOLE_0:def_4; h . E in rng h by A61, ENUMSET1:def_2; then t in h . E by A70, SETFAM_1:def_1; hence t in x by A24, A27, A30, A38, A73, XBOOLE_0:def_4; ::_thesis: verum end; then x = Intersect F by A62, XBOOLE_0:def_10; hence x in '/\' (G \ {A}) by A18, A17, A50, A47, A21, BVFUNC_2:def_1; ::_thesis: verum end; '/\' (G \ {A}) c= ((B '/\' C) '/\' D) '/\' E proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in '/\' (G \ {A}) or x in ((B '/\' C) '/\' D) '/\' E ) assume x in '/\' (G \ {A}) ; ::_thesis: x in ((B '/\' C) '/\' D) '/\' E then consider h being Function, F being Subset-Family of Y such that A74: dom h = G \ {A} and A75: rng h = F and A76: for d being set st d in G \ {A} holds h . d in d and A77: x = Intersect F and A78: x <> {} by BVFUNC_2:def_1; D in dom h by A18, A17, A74, ENUMSET1:def_2; then A79: h . D in rng h by FUNCT_1:def_3; set mbc = (h . B) /\ (h . C); A80: not x in {{}} by A78, TARSKI:def_1; E in G \ {A} by A18, A17, ENUMSET1:def_2; then A81: h . E in E by A76; D in G \ {A} by A18, A17, ENUMSET1:def_2; then A82: h . D in D by A76; C in G \ {A} by A18, A17, ENUMSET1:def_2; then A83: h . C in C by A76; E in dom h by A18, A17, A74, ENUMSET1:def_2; then A84: h . E in rng h by FUNCT_1:def_3; set mbcd = ((h . B) /\ (h . C)) /\ (h . D); B in dom h by A18, A17, A74, ENUMSET1:def_2; then A85: h . B in rng h by FUNCT_1:def_3; C in dom h by A18, A17, A74, ENUMSET1:def_2; then A86: h . C in rng h by FUNCT_1:def_3; A87: x c= (((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E) proof let m be set ; :: according to TARSKI:def_3 ::_thesis: ( not m in x or m in (((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E) ) assume m in x ; ::_thesis: m in (((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E) then A88: m in meet (rng h) by A75, A77, A85, SETFAM_1:def_9; then ( m in h . B & m in h . C ) by A85, A86, SETFAM_1:def_1; then A89: m in (h . B) /\ (h . C) by XBOOLE_0:def_4; m in h . D by A79, A88, SETFAM_1:def_1; then A90: m in ((h . B) /\ (h . C)) /\ (h . D) by A89, XBOOLE_0:def_4; m in h . E by A84, A88, SETFAM_1:def_1; hence m in (((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E) by A90, XBOOLE_0:def_4; ::_thesis: verum end; then ((h . B) /\ (h . C)) /\ (h . D) <> {} by A78; then A91: not ((h . B) /\ (h . C)) /\ (h . D) in {{}} by TARSKI:def_1; (h . B) /\ (h . C) <> {} by A78, A87; then A92: not (h . B) /\ (h . C) in {{}} by TARSKI:def_1; B in G \ {A} by A18, A17, ENUMSET1:def_2; then h . B in B by A76; then (h . B) /\ (h . C) in INTERSECTION (B,C) by A83, SETFAM_1:def_5; then (h . B) /\ (h . C) in (INTERSECTION (B,C)) \ {{}} by A92, XBOOLE_0:def_5; then (h . B) /\ (h . C) in B '/\' C by PARTIT1:def_4; then ((h . B) /\ (h . C)) /\ (h . D) in INTERSECTION ((B '/\' C),D) by A82, SETFAM_1:def_5; then ((h . B) /\ (h . C)) /\ (h . D) in (INTERSECTION ((B '/\' C),D)) \ {{}} by A91, XBOOLE_0:def_5; then A93: ((h . B) /\ (h . C)) /\ (h . D) in (B '/\' C) '/\' D by PARTIT1:def_4; (((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E) c= x proof let m be set ; :: according to TARSKI:def_3 ::_thesis: ( not m in (((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E) or m in x ) assume A94: m in (((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E) ; ::_thesis: m in x then A95: m in ((h . B) /\ (h . C)) /\ (h . D) by XBOOLE_0:def_4; then A96: m in (h . B) /\ (h . C) by XBOOLE_0:def_4; A97: rng h c= {(h . B),(h . C),(h . D),(h . E)} proof let u be set ; :: according to TARSKI:def_3 ::_thesis: ( not u in rng h or u in {(h . B),(h . C),(h . D),(h . E)} ) assume u in rng h ; ::_thesis: u in {(h . B),(h . C),(h . D),(h . E)} then consider x1 being set such that A98: x1 in dom h and A99: u = h . x1 by FUNCT_1:def_3; now__::_thesis:_(_(_x1_=_B_&_u_in_{(h_._B),(h_._C),(h_._D),(h_._E)}_)_or_(_x1_=_C_&_u_in_{(h_._B),(h_._C),(h_._D),(h_._E)}_)_or_(_x1_=_D_&_u_in_{(h_._B),(h_._C),(h_._D),(h_._E)}_)_or_(_x1_=_E_&_u_in_{(h_._B),(h_._C),(h_._D),(h_._E)}_)_) percases ( x1 = B or x1 = C or x1 = D or x1 = E ) by A15, A17, A74, A98, ENUMSET1:def_2; case x1 = B ; ::_thesis: u in {(h . B),(h . C),(h . D),(h . E)} hence u in {(h . B),(h . C),(h . D),(h . E)} by A99, ENUMSET1:def_2; ::_thesis: verum end; case x1 = C ; ::_thesis: u in {(h . B),(h . C),(h . D),(h . E)} hence u in {(h . B),(h . C),(h . D),(h . E)} by A99, ENUMSET1:def_2; ::_thesis: verum end; case x1 = D ; ::_thesis: u in {(h . B),(h . C),(h . D),(h . E)} hence u in {(h . B),(h . C),(h . D),(h . E)} by A99, ENUMSET1:def_2; ::_thesis: verum end; case x1 = E ; ::_thesis: u in {(h . B),(h . C),(h . D),(h . E)} hence u in {(h . B),(h . C),(h . D),(h . E)} by A99, ENUMSET1:def_2; ::_thesis: verum end; end; end; hence u in {(h . B),(h . C),(h . D),(h . E)} ; ::_thesis: verum end; for y being set st y in rng h holds m in y proof let y be set ; ::_thesis: ( y in rng h implies m in y ) assume A100: y in rng h ; ::_thesis: m in y now__::_thesis:_(_(_y_=_h_._B_&_m_in_y_)_or_(_y_=_h_._C_&_m_in_y_)_or_(_y_=_h_._D_&_m_in_y_)_or_(_y_=_h_._E_&_m_in_y_)_) percases ( y = h . B or y = h . C or y = h . D or y = h . E ) by A97, A100, ENUMSET1:def_2; case y = h . B ; ::_thesis: m in y hence m in y by A96, XBOOLE_0:def_4; ::_thesis: verum end; case y = h . C ; ::_thesis: m in y hence m in y by A96, XBOOLE_0:def_4; ::_thesis: verum end; case y = h . D ; ::_thesis: m in y hence m in y by A95, XBOOLE_0:def_4; ::_thesis: verum end; case y = h . E ; ::_thesis: m in y hence m in y by A94, XBOOLE_0:def_4; ::_thesis: verum end; end; end; hence m in y ; ::_thesis: verum end; then m in meet (rng h) by A85, SETFAM_1:def_1; hence m in x by A75, A77, A85, SETFAM_1:def_9; ::_thesis: verum end; then (((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E) = x by A87, XBOOLE_0:def_10; then x in INTERSECTION (((B '/\' C) '/\' D),E) by A81, A93, SETFAM_1:def_5; then x in (INTERSECTION (((B '/\' C) '/\' D),E)) \ {{}} by A80, XBOOLE_0:def_5; hence x in ((B '/\' C) '/\' D) '/\' E by PARTIT1:def_4; ::_thesis: verum end; then '/\' (G \ {A}) = ((B '/\' C) '/\' D) '/\' E by A19, XBOOLE_0:def_10; hence CompF (A,G) = ((B '/\' C) '/\' D) '/\' E by BVFUNC_2:def_7; ::_thesis: verum end; end; end; theorem Th22: :: BVFUNC14:22 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E being a_partition of Y st G = {A,B,C,D,E} & A <> B & B <> C & B <> D & B <> E holds CompF (B,G) = ((A '/\' C) '/\' D) '/\' E proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C, D, E being a_partition of Y st G = {A,B,C,D,E} & A <> B & B <> C & B <> D & B <> E holds CompF (B,G) = ((A '/\' C) '/\' D) '/\' E let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C, D, E being a_partition of Y st G = {A,B,C,D,E} & A <> B & B <> C & B <> D & B <> E holds CompF (B,G) = ((A '/\' C) '/\' D) '/\' E let A, B, C, D, E be a_partition of Y; ::_thesis: ( G = {A,B,C,D,E} & A <> B & B <> C & B <> D & B <> E implies CompF (B,G) = ((A '/\' C) '/\' D) '/\' E ) assume that A1: G = {A,B,C,D,E} and A2: ( A <> B & B <> C & B <> D & B <> E ) ; ::_thesis: CompF (B,G) = ((A '/\' C) '/\' D) '/\' E {A,B,C,D,E} = {A,B} \/ {C,D,E} by ENUMSET1:8; then G = {B,A,C,D,E} by A1, ENUMSET1:8; hence CompF (B,G) = ((A '/\' C) '/\' D) '/\' E by A2, Th21; ::_thesis: verum end; theorem Th23: :: BVFUNC14:23 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E being a_partition of Y st G = {A,B,C,D,E} & A <> C & B <> C & C <> D & C <> E holds CompF (C,G) = ((A '/\' B) '/\' D) '/\' E proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C, D, E being a_partition of Y st G = {A,B,C,D,E} & A <> C & B <> C & C <> D & C <> E holds CompF (C,G) = ((A '/\' B) '/\' D) '/\' E let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C, D, E being a_partition of Y st G = {A,B,C,D,E} & A <> C & B <> C & C <> D & C <> E holds CompF (C,G) = ((A '/\' B) '/\' D) '/\' E let A, B, C, D, E be a_partition of Y; ::_thesis: ( G = {A,B,C,D,E} & A <> C & B <> C & C <> D & C <> E implies CompF (C,G) = ((A '/\' B) '/\' D) '/\' E ) assume that A1: G = {A,B,C,D,E} and A2: ( A <> C & B <> C & C <> D & C <> E ) ; ::_thesis: CompF (C,G) = ((A '/\' B) '/\' D) '/\' E {A,B,C,D,E} = {A,B,C} \/ {D,E} by ENUMSET1:9; then {A,B,C,D,E} = ({A} \/ {B,C}) \/ {D,E} by ENUMSET1:2; then {A,B,C,D,E} = {A,C,B} \/ {D,E} by ENUMSET1:2; then {A,B,C,D,E} = ({A,C} \/ {B}) \/ {D,E} by ENUMSET1:3; then {A,B,C,D,E} = {C,A,B} \/ {D,E} by ENUMSET1:3; then G = {C,A,B,D,E} by A1, ENUMSET1:9; hence CompF (C,G) = ((A '/\' B) '/\' D) '/\' E by A2, Th21; ::_thesis: verum end; theorem Th24: :: BVFUNC14:24 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E being a_partition of Y st G = {A,B,C,D,E} & A <> D & B <> D & C <> D & D <> E holds CompF (D,G) = ((A '/\' B) '/\' C) '/\' E proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C, D, E being a_partition of Y st G = {A,B,C,D,E} & A <> D & B <> D & C <> D & D <> E holds CompF (D,G) = ((A '/\' B) '/\' C) '/\' E let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C, D, E being a_partition of Y st G = {A,B,C,D,E} & A <> D & B <> D & C <> D & D <> E holds CompF (D,G) = ((A '/\' B) '/\' C) '/\' E let A, B, C, D, E be a_partition of Y; ::_thesis: ( G = {A,B,C,D,E} & A <> D & B <> D & C <> D & D <> E implies CompF (D,G) = ((A '/\' B) '/\' C) '/\' E ) assume that A1: G = {A,B,C,D,E} and A2: ( A <> D & B <> D & C <> D & D <> E ) ; ::_thesis: CompF (D,G) = ((A '/\' B) '/\' C) '/\' E {A,B,C,D,E} = {A,B} \/ {C,D,E} by ENUMSET1:8; then {A,B,C,D,E} = {A,B} \/ ({C,D} \/ {E}) by ENUMSET1:3; then {A,B,C,D,E} = {A,B} \/ {D,C,E} by ENUMSET1:3; then G = {A,B,D,C,E} by A1, ENUMSET1:8; hence CompF (D,G) = ((A '/\' B) '/\' C) '/\' E by A2, Th23; ::_thesis: verum end; theorem :: BVFUNC14:25 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E being a_partition of Y st G = {A,B,C,D,E} & A <> E & B <> E & C <> E & D <> E holds CompF (E,G) = ((A '/\' B) '/\' C) '/\' D proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C, D, E being a_partition of Y st G = {A,B,C,D,E} & A <> E & B <> E & C <> E & D <> E holds CompF (E,G) = ((A '/\' B) '/\' C) '/\' D let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C, D, E being a_partition of Y st G = {A,B,C,D,E} & A <> E & B <> E & C <> E & D <> E holds CompF (E,G) = ((A '/\' B) '/\' C) '/\' D let A, B, C, D, E be a_partition of Y; ::_thesis: ( G = {A,B,C,D,E} & A <> E & B <> E & C <> E & D <> E implies CompF (E,G) = ((A '/\' B) '/\' C) '/\' D ) assume that A1: G = {A,B,C,D,E} and A2: ( A <> E & B <> E & C <> E & D <> E ) ; ::_thesis: CompF (E,G) = ((A '/\' B) '/\' C) '/\' D {A,B,C,D,E} = {A,B,C} \/ {D,E} by ENUMSET1:9; then G = {A,B,C,E,D} by A1, ENUMSET1:9; hence CompF (E,G) = ((A '/\' B) '/\' C) '/\' D by A2, Th24; ::_thesis: verum end; theorem Th26: :: BVFUNC14:26 for A, B, C, D, E being set for h being Function for A9, B9, C9, D9, E9 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 .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (A .--> A9) holds ( h . A = A9 & h . B = B9 & h . C = C9 & h . D = D9 & h . E = E9 ) proof let A, B, C, D, E be set ; ::_thesis: for h being Function for A9, B9, C9, D9, E9 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 .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (A .--> A9) holds ( h . A = A9 & h . B = B9 & h . C = C9 & h . D = D9 & h . E = E9 ) let h be Function; ::_thesis: for A9, B9, C9, D9, E9 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 .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (A .--> A9) holds ( h . A = A9 & h . B = B9 & h . C = C9 & h . D = D9 & h . E = E9 ) let A9, B9, C9, D9, E9 be set ; ::_thesis: ( A <> B & A <> C & A <> D & A <> E & B <> C & B <> D & B <> E & C <> D & C <> E & D <> E & h = ((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (A .--> A9) implies ( h . A = A9 & h . B = B9 & h . C = C9 & h . D = D9 & h . E = E9 ) ) assume that A1: A <> B and A2: A <> C and A3: A <> D and A4: A <> E and A5: B <> C and A6: B <> D and A7: B <> E and A8: C <> D and A9: C <> E and A10: D <> E and A11: h = ((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (A .--> A9) ; ::_thesis: ( h . A = A9 & h . B = B9 & h . C = C9 & h . D = D9 & h . E = E9 ) A12: dom (A .--> A9) = {A} by FUNCOP_1:13; then A in dom (A .--> A9) by TARSKI:def_1; then A13: h . A = (A .--> A9) . A by A11, FUNCT_4:13; not C in dom (A .--> A9) by A2, A12, TARSKI:def_1; then A14: h . C = ((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) . C by A11, FUNCT_4:11; A15: dom (D .--> D9) = {D} by FUNCOP_1:13; then not B in dom (D .--> D9) by A6, TARSKI:def_1; then A16: (((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) . B = ((B .--> B9) +* (C .--> C9)) . B by FUNCT_4:11; not E in dom (A .--> A9) by A4, A12, TARSKI:def_1; then A17: h . E = ((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) . E by A11, FUNCT_4:11; A18: dom (E .--> E9) = {E} by FUNCOP_1:13; then E in dom (E .--> E9) by TARSKI:def_1; then A19: h . E = (E .--> E9) . E by A17, FUNCT_4:13; not C in dom (D .--> D9) by A8, A15, TARSKI:def_1; then A20: (((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) . C = ((B .--> B9) +* (C .--> C9)) . C by FUNCT_4:11; not C in dom (E .--> E9) by A9, A18, TARSKI:def_1; then A21: h . C = (((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) . C by A14, FUNCT_4:11; A22: dom (C .--> C9) = {C} by FUNCOP_1:13; then C in dom (C .--> C9) by TARSKI:def_1; then A23: h . C = (C .--> C9) . C by A21, A20, FUNCT_4:13; not D in dom (A .--> A9) by A3, A12, TARSKI:def_1; then A24: h . D = ((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) . D by A11, FUNCT_4:11; not D in dom (E .--> E9) by A10, A18, TARSKI:def_1; then A25: h . D = (((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) . D by A24, FUNCT_4:11; D in dom (D .--> D9) by A15, TARSKI:def_1; then A26: h . D = (D .--> D9) . D by A25, FUNCT_4:13; not B in dom (A .--> A9) by A1, A12, TARSKI:def_1; then A27: h . B = ((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) . B by A11, FUNCT_4:11; not B in dom (E .--> E9) by A7, A18, TARSKI:def_1; then A28: h . B = (((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) . B by A27, FUNCT_4:11; not B in dom (C .--> C9) by A5, A22, TARSKI:def_1; then h . B = (B .--> B9) . B by A28, A16, FUNCT_4:11; hence ( h . A = A9 & h . B = B9 & h . C = C9 & h . D = D9 & h . E = E9 ) by A13, A23, A26, A19, FUNCOP_1:72; ::_thesis: verum end; theorem Th27: :: BVFUNC14:27 for A, B, C, D, E being set for h being Function for A9, B9, C9, D9, E9 being set st h = ((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (A .--> A9) holds dom h = {A,B,C,D,E} proof let A, B, C, D, E be set ; ::_thesis: for h being Function for A9, B9, C9, D9, E9 being set st h = ((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (A .--> A9) holds dom h = {A,B,C,D,E} let h be Function; ::_thesis: for A9, B9, C9, D9, E9 being set st h = ((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (A .--> A9) holds dom h = {A,B,C,D,E} let A9, B9, C9, D9, E9 be set ; ::_thesis: ( h = ((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (A .--> A9) implies dom h = {A,B,C,D,E} ) assume A1: h = ((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (A .--> A9) ; ::_thesis: dom h = {A,B,C,D,E} A2: ( dom (D .--> D9) = {D} & dom (E .--> E9) = {E} ) by FUNCOP_1:13; dom ((B .--> B9) +* (C .--> C9)) = (dom (B .--> B9)) \/ (dom (C .--> C9)) by FUNCT_4:def_1; then dom (((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) = ((dom (B .--> B9)) \/ (dom (C .--> C9))) \/ (dom (D .--> D9)) by FUNCT_4:def_1; then dom ((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) = (((dom (B .--> B9)) \/ (dom (C .--> C9))) \/ (dom (D .--> D9))) \/ (dom (E .--> E9)) by FUNCT_4:def_1; then A3: dom (((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (A .--> A9)) = ((((dom (B .--> B9)) \/ (dom (C .--> C9))) \/ (dom (D .--> D9))) \/ (dom (E .--> E9))) \/ (dom (A .--> A9)) by FUNCT_4:def_1; ( dom (B .--> B9) = {B} & dom (C .--> C9) = {C} ) by FUNCOP_1:13; then dom (((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (A .--> A9)) = {A} \/ ((({B} \/ {C}) \/ {D}) \/ {E}) by A3, A2, FUNCOP_1:13 .= {A} \/ (({B,C} \/ {D}) \/ {E}) by ENUMSET1:1 .= {A} \/ ({B,C,D} \/ {E}) by ENUMSET1:3 .= {A} \/ {B,C,D,E} by ENUMSET1:6 .= {A,B,C,D,E} by ENUMSET1:7 ; hence dom h = {A,B,C,D,E} by A1; ::_thesis: verum end; theorem Th28: :: BVFUNC14:28 for A, B, C, D, E being set for h being Function for A9, B9, C9, D9, E9 being set st h = ((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (A .--> A9) holds rng h = {(h . A),(h . B),(h . C),(h . D),(h . E)} proof let A, B, C, D, E be set ; ::_thesis: for h being Function for A9, B9, C9, D9, E9 being set st h = ((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (A .--> A9) holds rng h = {(h . A),(h . B),(h . C),(h . D),(h . E)} let h be Function; ::_thesis: for A9, B9, C9, D9, E9 being set st h = ((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (A .--> A9) holds rng h = {(h . A),(h . B),(h . C),(h . D),(h . E)} let A9, B9, C9, D9, E9 be set ; ::_thesis: ( h = ((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (A .--> A9) implies rng h = {(h . A),(h . B),(h . C),(h . D),(h . E)} ) assume h = ((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (A .--> A9) ; ::_thesis: rng h = {(h . A),(h . B),(h . C),(h . D),(h . E)} then A1: dom h = {A,B,C,D,E} by Th27; then A2: B in dom h by ENUMSET1:def_3; A3: D in dom h by A1, ENUMSET1:def_3; A4: C in dom h by A1, ENUMSET1:def_3; A5: rng h c= {(h . A),(h . B),(h . C),(h . D),(h . E)} proof let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in rng h or t in {(h . A),(h . B),(h . C),(h . D),(h . E)} ) assume t in rng h ; ::_thesis: t in {(h . A),(h . B),(h . C),(h . D),(h . E)} then consider x1 being set such that A6: x1 in dom h and A7: t = h . x1 by FUNCT_1:def_3; now__::_thesis:_(_(_x1_=_A_&_t_in_{(h_._A),(h_._B),(h_._C),(h_._D),(h_._E)}_)_or_(_x1_=_B_&_t_in_{(h_._A),(h_._B),(h_._C),(h_._D),(h_._E)}_)_or_(_x1_=_C_&_t_in_{(h_._A),(h_._B),(h_._C),(h_._D),(h_._E)}_)_or_(_x1_=_D_&_t_in_{(h_._A),(h_._B),(h_._C),(h_._D),(h_._E)}_)_or_(_x1_=_E_&_t_in_{(h_._A),(h_._B),(h_._C),(h_._D),(h_._E)}_)_) percases ( x1 = A or x1 = B or x1 = C or x1 = D or x1 = E ) by A1, A6, ENUMSET1:def_3; case x1 = A ; ::_thesis: t in {(h . A),(h . B),(h . C),(h . D),(h . E)} hence t in {(h . A),(h . B),(h . C),(h . D),(h . E)} by A7, ENUMSET1:def_3; ::_thesis: verum end; case x1 = B ; ::_thesis: t in {(h . A),(h . B),(h . C),(h . D),(h . E)} hence t in {(h . A),(h . B),(h . C),(h . D),(h . E)} by A7, ENUMSET1:def_3; ::_thesis: verum end; case x1 = C ; ::_thesis: t in {(h . A),(h . B),(h . C),(h . D),(h . E)} hence t in {(h . A),(h . B),(h . C),(h . D),(h . E)} by A7, ENUMSET1:def_3; ::_thesis: verum end; case x1 = D ; ::_thesis: t in {(h . A),(h . B),(h . C),(h . D),(h . E)} hence t in {(h . A),(h . B),(h . C),(h . D),(h . E)} by A7, ENUMSET1:def_3; ::_thesis: verum end; case x1 = E ; ::_thesis: t in {(h . A),(h . B),(h . C),(h . D),(h . E)} hence t in {(h . A),(h . B),(h . C),(h . D),(h . E)} by A7, ENUMSET1:def_3; ::_thesis: verum end; end; end; hence t in {(h . A),(h . B),(h . C),(h . D),(h . E)} ; ::_thesis: verum end; A8: E in dom h by A1, ENUMSET1:def_3; A9: A in dom h by A1, ENUMSET1:def_3; {(h . A),(h . B),(h . C),(h . D),(h . E)} c= rng h proof let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in {(h . A),(h . B),(h . C),(h . D),(h . E)} or t in rng h ) assume A10: t in {(h . A),(h . B),(h . C),(h . D),(h . E)} ; ::_thesis: t in rng h now__::_thesis:_(_(_t_=_h_._A_&_t_in_rng_h_)_or_(_t_=_h_._B_&_t_in_rng_h_)_or_(_t_=_h_._C_&_t_in_rng_h_)_or_(_t_=_h_._D_&_t_in_rng_h_)_or_(_t_=_h_._E_&_t_in_rng_h_)_) percases ( t = h . A or t = h . B or t = h . C or t = h . D or t = h . E ) by A10, ENUMSET1:def_3; case t = h . A ; ::_thesis: t in rng h hence t in rng h by A9, FUNCT_1:def_3; ::_thesis: verum end; case t = h . B ; ::_thesis: t in rng h hence t in rng h by A2, FUNCT_1:def_3; ::_thesis: verum end; case t = h . C ; ::_thesis: t in rng h hence t in rng h by A4, FUNCT_1:def_3; ::_thesis: verum end; case t = h . D ; ::_thesis: t in rng h hence t in rng h by A3, FUNCT_1:def_3; ::_thesis: verum end; case t = h . E ; ::_thesis: t in rng h hence t in rng h by A8, FUNCT_1:def_3; ::_thesis: verum end; end; end; hence t in rng h ; ::_thesis: verum end; hence rng h = {(h . A),(h . B),(h . C),(h . D),(h . E)} by A5, XBOOLE_0:def_10; ::_thesis: verum end; theorem :: BVFUNC14:29 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E being a_partition of Y for z, u being Element of Y for 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 Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C, D, E being a_partition of Y for z, u being Element of Y for 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) let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C, D, E being a_partition of Y for z, u being Element of Y for 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) let A, B, C, D, E be a_partition of Y; ::_thesis: for z, u being Element of Y for 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) let z, u be Element of Y; ::_thesis: for 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) let h be Function; ::_thesis: ( 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 EqClass (u,(((B '/\' C) '/\' D) '/\' E)) meets EqClass (z,A) ) assume that A1: G is independent and A2: G = {A,B,C,D,E} and A3: ( A <> B & A <> C & A <> D & A <> E & B <> C & B <> D & B <> E & C <> D & C <> E & D <> E ) ; ::_thesis: EqClass (u,(((B '/\' C) '/\' D) '/\' E)) meets EqClass (z,A) set h = ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A))); A4: (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . B = EqClass (u,B) by A3, Th26; A5: (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . D = EqClass (u,D) by A3, Th26; A6: (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . C = EqClass (u,C) by A3, Th26; A7: (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . E = EqClass (u,E) by A3, Th26; A8: rng (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) = {((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . A),((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . B),((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . C),((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . D),((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . E)} by Th28; rng (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) c= bool Y proof let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in rng (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) or t in bool Y ) assume A9: t in rng (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) ; ::_thesis: t in bool Y now__::_thesis:_(_(_t_=_(((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(D_.-->_(EqClass_(u,D))))_+*_(E_.-->_(EqClass_(u,E))))_+*_(A_.-->_(EqClass_(z,A))))_._A_&_t_in_bool_Y_)_or_(_t_=_(((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(D_.-->_(EqClass_(u,D))))_+*_(E_.-->_(EqClass_(u,E))))_+*_(A_.-->_(EqClass_(z,A))))_._B_&_t_in_bool_Y_)_or_(_t_=_(((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(D_.-->_(EqClass_(u,D))))_+*_(E_.-->_(EqClass_(u,E))))_+*_(A_.-->_(EqClass_(z,A))))_._C_&_t_in_bool_Y_)_or_(_t_=_(((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(D_.-->_(EqClass_(u,D))))_+*_(E_.-->_(EqClass_(u,E))))_+*_(A_.-->_(EqClass_(z,A))))_._D_&_t_in_bool_Y_)_or_(_t_=_(((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(D_.-->_(EqClass_(u,D))))_+*_(E_.-->_(EqClass_(u,E))))_+*_(A_.-->_(EqClass_(z,A))))_._E_&_t_in_bool_Y_)_) percases ( t = (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . A or t = (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . B or t = (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . C or t = (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . D or t = (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . E ) by A8, A9, ENUMSET1:def_3; case t = (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . A ; ::_thesis: t in bool Y then t = EqClass (z,A) by A3, Th26; hence t in bool Y ; ::_thesis: verum end; case t = (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . B ; ::_thesis: t in bool Y hence t in bool Y by A4; ::_thesis: verum end; case t = (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . C ; ::_thesis: t in bool Y hence t in bool Y by A6; ::_thesis: verum end; case t = (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . D ; ::_thesis: t in bool Y hence t in bool Y by A5; ::_thesis: verum end; case t = (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . E ; ::_thesis: t in bool Y hence t in bool Y by A7; ::_thesis: verum end; end; end; hence t in bool Y ; ::_thesis: verum end; then reconsider FF = rng (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) as Subset-Family of Y ; A10: dom (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) = G by A2, Th27; for d being set st d in G holds (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . d in d proof let d be set ; ::_thesis: ( d in G implies (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . d in d ) assume A11: d in G ; ::_thesis: (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . d in d now__::_thesis:_(_(_d_=_A_&_(((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(D_.-->_(EqClass_(u,D))))_+*_(E_.-->_(EqClass_(u,E))))_+*_(A_.-->_(EqClass_(z,A))))_._d_in_d_)_or_(_d_=_B_&_(((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(D_.-->_(EqClass_(u,D))))_+*_(E_.-->_(EqClass_(u,E))))_+*_(A_.-->_(EqClass_(z,A))))_._d_in_d_)_or_(_d_=_C_&_(((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(D_.-->_(EqClass_(u,D))))_+*_(E_.-->_(EqClass_(u,E))))_+*_(A_.-->_(EqClass_(z,A))))_._d_in_d_)_or_(_d_=_D_&_(((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(D_.-->_(EqClass_(u,D))))_+*_(E_.-->_(EqClass_(u,E))))_+*_(A_.-->_(EqClass_(z,A))))_._d_in_d_)_or_(_d_=_E_&_(((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(D_.-->_(EqClass_(u,D))))_+*_(E_.-->_(EqClass_(u,E))))_+*_(A_.-->_(EqClass_(z,A))))_._d_in_d_)_) percases ( d = A or d = B or d = C or d = D or d = E ) by A2, A11, ENUMSET1:def_3; caseA12: d = A ; ::_thesis: (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . d in d (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . A = EqClass (z,A) by A3, Th26; hence (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . d in d by A12; ::_thesis: verum end; caseA13: d = B ; ::_thesis: (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . d in d (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . B = EqClass (u,B) by A3, Th26; hence (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . d in d by A13; ::_thesis: verum end; caseA14: d = C ; ::_thesis: (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . d in d (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . C = EqClass (u,C) by A3, Th26; hence (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . d in d by A14; ::_thesis: verum end; caseA15: d = D ; ::_thesis: (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . d in d (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . D = EqClass (u,D) by A3, Th26; hence (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . d in d by A15; ::_thesis: verum end; caseA16: d = E ; ::_thesis: (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . d in d (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . E = EqClass (u,E) by A3, Th26; hence (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . d in d by A16; ::_thesis: verum end; end; end; hence (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . d in d ; ::_thesis: verum end; then Intersect FF <> {} by A1, A10, BVFUNC_2:def_5; then consider m being set such that A17: m in Intersect FF by XBOOLE_0:def_1; A in dom (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) by A2, A10, ENUMSET1:def_3; then A18: (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . A in rng (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then A19: m in meet FF by A17, SETFAM_1:def_9; then A20: m in (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . A by A18, SETFAM_1:def_1; D in dom (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) by A2, A10, ENUMSET1:def_3; then (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . D in rng (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then A21: m in (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . D by A19, SETFAM_1:def_1; C in dom (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) by A2, A10, ENUMSET1:def_3; then (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . C in rng (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then A22: m in (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . C by A19, SETFAM_1:def_1; B in dom (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) by A2, A10, ENUMSET1:def_3; then (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . B in rng (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then m in (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . B by A19, SETFAM_1:def_1; then m in (EqClass (u,B)) /\ (EqClass (u,C)) by A4, A6, A22, XBOOLE_0:def_4; then A23: m in ((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D)) by A5, A21, XBOOLE_0:def_4; E in dom (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) by A2, A10, ENUMSET1:def_3; then (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . E in rng (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then m in (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . E by A19, SETFAM_1:def_1; then A24: m in (((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E)) by A7, A23, XBOOLE_0:def_4; set GG = EqClass (u,(((B '/\' C) '/\' D) '/\' E)); EqClass (u,(((B '/\' C) '/\' D) '/\' E)) = (EqClass (u,((B '/\' C) '/\' D))) /\ (EqClass (u,E)) by Th1; then A25: EqClass (u,(((B '/\' C) '/\' D) '/\' E)) = ((EqClass (u,(B '/\' C))) /\ (EqClass (u,D))) /\ (EqClass (u,E)) by Th1; (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . A = EqClass (z,A) by A3, Th26; then m in ((((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E))) /\ (EqClass (z,A)) by A20, A24, XBOOLE_0:def_4; then (((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E)) meets EqClass (z,A) by XBOOLE_0:4; hence EqClass (u,(((B '/\' C) '/\' D) '/\' E)) meets EqClass (z,A) by A25, Th1; ::_thesis: verum end; theorem :: BVFUNC14:30 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E being a_partition of Y for 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 Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C, D, E being a_partition of Y for 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))) let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C, D, E being a_partition of Y for 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))) let A, B, C, D, E be a_partition of Y; ::_thesis: for 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))) let z, u be Element of Y; ::_thesis: ( 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)) implies EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G))) ) assume that A1: G is independent and A2: G = {A,B,C,D,E} and A3: A <> B and A4: ( A <> C & A <> D & A <> E ) and A5: ( B <> C & B <> D & B <> E ) and A6: ( C <> D & C <> E & D <> E ) and A7: EqClass (z,((C '/\' D) '/\' E)) = EqClass (u,((C '/\' D) '/\' E)) ; ::_thesis: EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G))) set h = ((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A))); A8: (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . B = EqClass (u,B) by A3, A4, A5, A6, Th26; A9: (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . E = EqClass (u,E) by A3, A4, A5, A6, Th26; A10: (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . D = EqClass (u,D) by A3, A4, A5, A6, Th26; A11: (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . C = EqClass (u,C) by A3, A4, A5, A6, Th26; A12: rng (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) = {((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . A),((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . B),((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . C),((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . D),((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . E)} by Th28; rng (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) c= bool Y proof let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in rng (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) or t in bool Y ) assume A13: t in rng (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) ; ::_thesis: t in bool Y now__::_thesis:_(_(_t_=_(((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(D_.-->_(EqClass_(u,D))))_+*_(E_.-->_(EqClass_(u,E))))_+*_(A_.-->_(EqClass_(z,A))))_._A_&_t_in_bool_Y_)_or_(_t_=_(((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(D_.-->_(EqClass_(u,D))))_+*_(E_.-->_(EqClass_(u,E))))_+*_(A_.-->_(EqClass_(z,A))))_._B_&_t_in_bool_Y_)_or_(_t_=_(((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(D_.-->_(EqClass_(u,D))))_+*_(E_.-->_(EqClass_(u,E))))_+*_(A_.-->_(EqClass_(z,A))))_._C_&_t_in_bool_Y_)_or_(_t_=_(((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(D_.-->_(EqClass_(u,D))))_+*_(E_.-->_(EqClass_(u,E))))_+*_(A_.-->_(EqClass_(z,A))))_._D_&_t_in_bool_Y_)_or_(_t_=_(((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(D_.-->_(EqClass_(u,D))))_+*_(E_.-->_(EqClass_(u,E))))_+*_(A_.-->_(EqClass_(z,A))))_._E_&_t_in_bool_Y_)_) percases ( t = (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . A or t = (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . B or t = (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . C or t = (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . D or t = (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . E ) by A12, A13, ENUMSET1:def_3; case t = (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . A ; ::_thesis: t in bool Y then t = EqClass (z,A) by A3, A4, A5, A6, Th26; hence t in bool Y ; ::_thesis: verum end; case t = (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . B ; ::_thesis: t in bool Y hence t in bool Y by A8; ::_thesis: verum end; case t = (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . C ; ::_thesis: t in bool Y hence t in bool Y by A11; ::_thesis: verum end; case t = (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . D ; ::_thesis: t in bool Y hence t in bool Y by A10; ::_thesis: verum end; case t = (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . E ; ::_thesis: t in bool Y hence t in bool Y by A9; ::_thesis: verum end; end; end; hence t in bool Y ; ::_thesis: verum end; then reconsider FF = rng (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) as Subset-Family of Y ; A14: dom (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) = G by A2, Th27; for d being set st d in G holds (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . d in d proof let d be set ; ::_thesis: ( d in G implies (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . d in d ) assume A15: d in G ; ::_thesis: (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . d in d now__::_thesis:_(_(_d_=_A_&_(((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(D_.-->_(EqClass_(u,D))))_+*_(E_.-->_(EqClass_(u,E))))_+*_(A_.-->_(EqClass_(z,A))))_._d_in_d_)_or_(_d_=_B_&_(((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(D_.-->_(EqClass_(u,D))))_+*_(E_.-->_(EqClass_(u,E))))_+*_(A_.-->_(EqClass_(z,A))))_._d_in_d_)_or_(_d_=_C_&_(((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(D_.-->_(EqClass_(u,D))))_+*_(E_.-->_(EqClass_(u,E))))_+*_(A_.-->_(EqClass_(z,A))))_._d_in_d_)_or_(_d_=_D_&_(((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(D_.-->_(EqClass_(u,D))))_+*_(E_.-->_(EqClass_(u,E))))_+*_(A_.-->_(EqClass_(z,A))))_._d_in_d_)_or_(_d_=_E_&_(((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(D_.-->_(EqClass_(u,D))))_+*_(E_.-->_(EqClass_(u,E))))_+*_(A_.-->_(EqClass_(z,A))))_._d_in_d_)_) percases ( d = A or d = B or d = C or d = D or d = E ) by A2, A15, ENUMSET1:def_3; caseA16: d = A ; ::_thesis: (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . d in d (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . A = EqClass (z,A) by A3, A4, A5, A6, Th26; hence (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . d in d by A16; ::_thesis: verum end; caseA17: d = B ; ::_thesis: (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . d in d (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . B = EqClass (u,B) by A3, A4, A5, A6, Th26; hence (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . d in d by A17; ::_thesis: verum end; caseA18: d = C ; ::_thesis: (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . d in d (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . C = EqClass (u,C) by A3, A4, A5, A6, Th26; hence (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . d in d by A18; ::_thesis: verum end; caseA19: d = D ; ::_thesis: (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . d in d (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . D = EqClass (u,D) by A3, A4, A5, A6, Th26; hence (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . d in d by A19; ::_thesis: verum end; caseA20: d = E ; ::_thesis: (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . d in d (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . E = EqClass (u,E) by A3, A4, A5, A6, Th26; hence (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . d in d by A20; ::_thesis: verum end; end; end; hence (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . d in d ; ::_thesis: verum end; then Intersect FF <> {} by A1, A14, BVFUNC_2:def_5; then consider m being set such that A21: m in Intersect FF by XBOOLE_0:def_1; A in dom (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) by A2, A14, ENUMSET1:def_3; then A22: (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . A in rng (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then A23: m in meet FF by A21, SETFAM_1:def_9; then A24: m in (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . A by A22, SETFAM_1:def_1; D in dom (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) by A2, A14, ENUMSET1:def_3; then (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . D in rng (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then A25: m in (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . D by A23, SETFAM_1:def_1; C in dom (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) by A2, A14, ENUMSET1:def_3; then (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . C in rng (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then A26: m in (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . C by A23, SETFAM_1:def_1; B in dom (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) by A2, A14, ENUMSET1:def_3; then (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . B in rng (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then m in (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . B by A23, SETFAM_1:def_1; then m in (EqClass (u,B)) /\ (EqClass (u,C)) by A8, A11, A26, XBOOLE_0:def_4; then A27: m in ((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D)) by A10, A25, XBOOLE_0:def_4; E in dom (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) by A2, A14, ENUMSET1:def_3; then (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . E in rng (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then m in (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . E by A23, SETFAM_1:def_1; then A28: m in (((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E)) by A9, A27, XBOOLE_0:def_4; set GG = EqClass (u,(((B '/\' C) '/\' D) '/\' E)); set I = EqClass (z,A); EqClass (u,(((B '/\' C) '/\' D) '/\' E)) = (EqClass (u,((B '/\' C) '/\' D))) /\ (EqClass (u,E)) by Th1; then A29: EqClass (u,(((B '/\' C) '/\' D) '/\' E)) = ((EqClass (u,(B '/\' C))) /\ (EqClass (u,D))) /\ (EqClass (u,E)) by Th1; (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (A .--> (EqClass (z,A)))) . A = EqClass (z,A) by A3, A4, A5, A6, Th26; then m in ((((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E))) /\ (EqClass (z,A)) by A24, A28, XBOOLE_0:def_4; then (EqClass (u,(((B '/\' C) '/\' D) '/\' E))) /\ (EqClass (z,A)) <> {} by A29, Th1; then consider p being set such that A30: p in (EqClass (u,(((B '/\' C) '/\' D) '/\' E))) /\ (EqClass (z,A)) by XBOOLE_0:def_1; reconsider p = p as Element of Y by A30; set K = EqClass (p,((C '/\' D) '/\' E)); A31: p in EqClass (u,(((B '/\' C) '/\' D) '/\' E)) by A30, XBOOLE_0:def_4; A32: z in EqClass (z,A) by EQREL_1:def_6; set L = EqClass (z,((C '/\' D) '/\' E)); A33: p in EqClass (p,((C '/\' D) '/\' E)) by EQREL_1:def_6; EqClass (u,(((B '/\' C) '/\' D) '/\' E)) = EqClass (u,((B '/\' (C '/\' D)) '/\' E)) by PARTIT1:14; then EqClass (u,(((B '/\' C) '/\' D) '/\' E)) = EqClass (u,(B '/\' ((C '/\' D) '/\' E))) by PARTIT1:14; then EqClass (u,(((B '/\' C) '/\' D) '/\' E)) c= EqClass (z,((C '/\' D) '/\' E)) by A7, BVFUNC11:3; then EqClass (p,((C '/\' D) '/\' E)) meets EqClass (z,((C '/\' D) '/\' E)) by A31, A33, XBOOLE_0:3; then EqClass (p,((C '/\' D) '/\' E)) = EqClass (z,((C '/\' D) '/\' E)) by EQREL_1:41; then z in EqClass (p,((C '/\' D) '/\' E)) by EQREL_1:def_6; then A34: z in (EqClass (z,A)) /\ (EqClass (p,((C '/\' D) '/\' E))) by A32, XBOOLE_0:def_4; set H = EqClass (z,(CompF (B,G))); A '/\' ((C '/\' D) '/\' E) = (A '/\' (C '/\' D)) '/\' E by PARTIT1:14; then A35: A '/\' ((C '/\' D) '/\' E) = ((A '/\' C) '/\' D) '/\' E by PARTIT1:14; A36: ( p in EqClass (p,((C '/\' D) '/\' E)) & p in EqClass (z,A) ) by A30, EQREL_1:def_6, XBOOLE_0:def_4; then p in (EqClass (z,A)) /\ (EqClass (p,((C '/\' D) '/\' E))) by XBOOLE_0:def_4; then ( (EqClass (z,A)) /\ (EqClass (p,((C '/\' D) '/\' E))) in INTERSECTION (A,((C '/\' D) '/\' E)) & not (EqClass (z,A)) /\ (EqClass (p,((C '/\' D) '/\' E))) in {{}} ) by SETFAM_1:def_5, TARSKI:def_1; then A37: (EqClass (z,A)) /\ (EqClass (p,((C '/\' D) '/\' E))) in (INTERSECTION (A,((C '/\' D) '/\' E))) \ {{}} by XBOOLE_0:def_5; CompF (B,G) = ((A '/\' C) '/\' D) '/\' E by A2, A3, A5, Th22; then (EqClass (z,A)) /\ (EqClass (p,((C '/\' D) '/\' E))) in CompF (B,G) by A37, A35, PARTIT1:def_4; then A38: ( (EqClass (z,A)) /\ (EqClass (p,((C '/\' D) '/\' E))) = EqClass (z,(CompF (B,G))) or (EqClass (z,A)) /\ (EqClass (p,((C '/\' D) '/\' E))) misses EqClass (z,(CompF (B,G))) ) by EQREL_1:def_4; z in EqClass (z,(CompF (B,G))) by EQREL_1:def_6; then p in EqClass (z,(CompF (B,G))) by A36, A34, A38, XBOOLE_0:3, XBOOLE_0:def_4; then p in (EqClass (u,(((B '/\' C) '/\' D) '/\' E))) /\ (EqClass (z,(CompF (B,G)))) by A31, XBOOLE_0:def_4; then EqClass (u,(((B '/\' C) '/\' D) '/\' E)) meets EqClass (z,(CompF (B,G))) by XBOOLE_0:4; hence EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G))) by A2, A3, A4, Th21; ::_thesis: verum end; theorem Th31: :: BVFUNC14:31 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F being a_partition of Y st G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F holds CompF (A,G) = (((B '/\' C) '/\' D) '/\' E) '/\' F proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F being a_partition of Y st G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F holds CompF (A,G) = (((B '/\' C) '/\' D) '/\' E) '/\' F let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C, D, E, F being a_partition of Y st G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F holds CompF (A,G) = (((B '/\' C) '/\' D) '/\' E) '/\' F let A, B, C, D, E, F be a_partition of Y; ::_thesis: ( G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F implies CompF (A,G) = (((B '/\' C) '/\' D) '/\' E) '/\' F ) assume that A1: G = {A,B,C,D,E,F} and A2: A <> B and A3: A <> C and A4: ( A <> D & A <> E ) and A5: A <> F and A6: ( B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F ) ; ::_thesis: CompF (A,G) = (((B '/\' C) '/\' D) '/\' E) '/\' F A7: G \ {A} = ({A} \/ {B,C,D,E,F}) \ {A} by A1, ENUMSET1:11 .= ({A} \ {A}) \/ ({B,C,D,E,F} \ {A}) by XBOOLE_1:42 ; A8: not F in {A} by A5, TARSKI:def_1; A9: ( not D in {A} & not E in {A} ) by A4, TARSKI:def_1; A10: not C in {A} by A3, TARSKI:def_1; A11: not B in {A} by A2, TARSKI:def_1; A in {A} by TARSKI:def_1; then A12: {A} \ {A} = {} by ZFMISC_1:60; A13: {B,C,D,E,F} \ {A} = ({B} \/ {C,D,E,F}) \ {A} by ENUMSET1:7 .= ({B} \ {A}) \/ ({C,D,E,F} \ {A}) by XBOOLE_1:42 .= {B} \/ ({C,D,E,F} \ {A}) by A11, ZFMISC_1:59 .= {B} \/ (({C} \/ {D,E,F}) \ {A}) by ENUMSET1:4 .= {B} \/ (({C} \ {A}) \/ ({D,E,F} \ {A})) by XBOOLE_1:42 .= {B} \/ (({C} \ {A}) \/ (({D,E} \/ {F}) \ {A})) by ENUMSET1:3 .= {B} \/ (({C} \ {A}) \/ (({D,E} \ {A}) \/ ({F} \ {A}))) by XBOOLE_1:42 .= {B} \/ (({C} \ {A}) \/ ({D,E} \/ ({F} \ {A}))) by A9, ZFMISC_1:63 .= {B} \/ (({C} \ {A}) \/ ({D,E} \/ {F})) by A8, ZFMISC_1:59 .= {B} \/ ({C} \/ ({D,E} \/ {F})) by A10, ZFMISC_1:59 .= {B} \/ ({C} \/ {D,E,F}) by ENUMSET1:3 .= {B} \/ {C,D,E,F} by ENUMSET1:4 .= {B,C,D,E,F} by ENUMSET1:7 ; A14: (((B '/\' C) '/\' D) '/\' E) '/\' F c= '/\' (G \ {A}) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (((B '/\' C) '/\' D) '/\' E) '/\' F or x in '/\' (G \ {A}) ) assume A15: x in (((B '/\' C) '/\' D) '/\' E) '/\' F ; ::_thesis: x in '/\' (G \ {A}) then A16: x <> {} by EQREL_1:def_4; x in (INTERSECTION ((((B '/\' C) '/\' D) '/\' E),F)) \ {{}} by A15, PARTIT1:def_4; then consider bcde, f being set such that A17: bcde in ((B '/\' C) '/\' D) '/\' E and A18: f in F and A19: x = bcde /\ f by SETFAM_1:def_5; bcde in (INTERSECTION (((B '/\' C) '/\' D),E)) \ {{}} by A17, PARTIT1:def_4; then consider bcd, e being set such that A20: bcd in (B '/\' C) '/\' D and A21: e in E and A22: bcde = bcd /\ e by SETFAM_1:def_5; bcd in (INTERSECTION ((B '/\' C),D)) \ {{}} by A20, PARTIT1:def_4; then consider bc, d being set such that A23: bc in B '/\' C and A24: d in D and A25: bcd = bc /\ d by SETFAM_1:def_5; bc in (INTERSECTION (B,C)) \ {{}} by A23, PARTIT1:def_4; then consider b, c being set such that A26: b in B and A27: c in C and A28: bc = b /\ c by SETFAM_1:def_5; set h = ((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f); A29: (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . B = b by A6, Th26; A30: (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . E = e by A6, Th26; A31: (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . F = f by A6, Th26; A32: dom (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) = {F,B,C,D,E} by Th27 .= {F} \/ {B,C,D,E} by ENUMSET1:7 .= {B,C,D,E,F} by ENUMSET1:10 ; then A33: C in dom (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) by ENUMSET1:def_3; A34: F in dom (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) by A32, ENUMSET1:def_3; A35: E in dom (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) by A32, ENUMSET1:def_3; A36: (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . C = c by A6, Th26; A37: rng (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) c= {((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . D),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . B),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . C),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . E),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . F)} proof let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in rng (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) or t in {((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . D),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . B),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . C),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . E),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . F)} ) assume t in rng (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) ; ::_thesis: t in {((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . D),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . B),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . C),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . E),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . F)} then consider x1 being set such that A38: x1 in dom (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) and A39: t = (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . x1 by FUNCT_1:def_3; now__::_thesis:_(_(_x1_=_D_&_t_in_{((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_._D),((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_._B),((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_._C),((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_._E),((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_._F)}_)_or_(_x1_=_B_&_t_in_{((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_._D),((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_._B),((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_._C),((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_._E),((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_._F)}_)_or_(_x1_=_C_&_t_in_{((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_._D),((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_._B),((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_._C),((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_._E),((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_._F)}_)_or_(_x1_=_E_&_t_in_{((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_._D),((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_._B),((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_._C),((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_._E),((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_._F)}_)_or_(_x1_=_F_&_t_in_{((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_._D),((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_._B),((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_._C),((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_._E),((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_._F)}_)_) percases ( x1 = D or x1 = B or x1 = C or x1 = E or x1 = F ) by A32, A38, ENUMSET1:def_3; case x1 = D ; ::_thesis: t in {((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . D),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . B),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . C),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . E),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . F)} hence t in {((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . D),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . B),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . C),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . E),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . F)} by A39, ENUMSET1:def_3; ::_thesis: verum end; case x1 = B ; ::_thesis: t in {((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . D),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . B),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . C),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . E),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . F)} hence t in {((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . D),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . B),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . C),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . E),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . F)} by A39, ENUMSET1:def_3; ::_thesis: verum end; case x1 = C ; ::_thesis: t in {((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . D),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . B),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . C),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . E),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . F)} hence t in {((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . D),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . B),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . C),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . E),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . F)} by A39, ENUMSET1:def_3; ::_thesis: verum end; case x1 = E ; ::_thesis: t in {((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . D),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . B),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . C),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . E),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . F)} hence t in {((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . D),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . B),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . C),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . E),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . F)} by A39, ENUMSET1:def_3; ::_thesis: verum end; case x1 = F ; ::_thesis: t in {((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . D),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . B),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . C),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . E),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . F)} hence t in {((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . D),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . B),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . C),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . E),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . F)} by A39, ENUMSET1:def_3; ::_thesis: verum end; end; end; hence t in {((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . D),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . B),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . C),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . E),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . F)} ; ::_thesis: verum end; A40: (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . D = d by A6, Th26; rng (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) c= bool Y proof let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in rng (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) or t in bool Y ) assume A41: t in rng (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) ; ::_thesis: t in bool Y now__::_thesis:_(_(_t_=_(((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_._D_&_t_in_bool_Y_)_or_(_t_=_(((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_._B_&_t_in_bool_Y_)_or_(_t_=_(((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_._C_&_t_in_bool_Y_)_or_(_t_=_(((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_._E_&_t_in_bool_Y_)_or_(_t_=_(((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_._F_&_t_in_bool_Y_)_) percases ( t = (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . D or t = (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . B or t = (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . C or t = (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . E or t = (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . F ) by A37, A41, ENUMSET1:def_3; case t = (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . D ; ::_thesis: t in bool Y hence t in bool Y by A24, A40; ::_thesis: verum end; case t = (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . B ; ::_thesis: t in bool Y hence t in bool Y by A26, A29; ::_thesis: verum end; case t = (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . C ; ::_thesis: t in bool Y hence t in bool Y by A27, A36; ::_thesis: verum end; case t = (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . E ; ::_thesis: t in bool Y hence t in bool Y by A21, A30; ::_thesis: verum end; case t = (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . F ; ::_thesis: t in bool Y hence t in bool Y by A18, A31; ::_thesis: verum end; end; end; hence t in bool Y ; ::_thesis: verum end; then reconsider FF = rng (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) as Subset-Family of Y ; A42: D in dom (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) by A32, ENUMSET1:def_3; then (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . D in rng (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) by FUNCT_1:def_3; then A43: Intersect FF = meet (rng (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f))) by SETFAM_1:def_9; A44: B in dom (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) by A32, ENUMSET1:def_3; {((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . D),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . B),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . C),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . E),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . F)} c= rng (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) proof let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in {((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . D),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . B),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . C),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . E),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . F)} or t in rng (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) ) assume A45: t in {((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . D),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . B),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . C),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . E),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . F)} ; ::_thesis: t in rng (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) now__::_thesis:_(_(_t_=_(((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_._D_&_t_in_rng_(((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_)_or_(_t_=_(((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_._B_&_t_in_rng_(((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_)_or_(_t_=_(((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_._C_&_t_in_rng_(((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_)_or_(_t_=_(((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_._E_&_t_in_rng_(((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_)_or_(_t_=_(((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_._F_&_t_in_rng_(((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_)_) percases ( t = (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . D or t = (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . B or t = (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . C or t = (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . E or t = (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . F ) by A45, ENUMSET1:def_3; case t = (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . D ; ::_thesis: t in rng (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) hence t in rng (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) by A42, FUNCT_1:def_3; ::_thesis: verum end; case t = (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . B ; ::_thesis: t in rng (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) hence t in rng (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) by A44, FUNCT_1:def_3; ::_thesis: verum end; case t = (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . C ; ::_thesis: t in rng (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) hence t in rng (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) by A33, FUNCT_1:def_3; ::_thesis: verum end; case t = (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . E ; ::_thesis: t in rng (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) hence t in rng (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) by A35, FUNCT_1:def_3; ::_thesis: verum end; case t = (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . F ; ::_thesis: t in rng (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) hence t in rng (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) by A34, FUNCT_1:def_3; ::_thesis: verum end; end; end; hence t in rng (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) ; ::_thesis: verum end; then A46: rng (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) = {((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . D),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . B),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . C),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . E),((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . F)} by A37, XBOOLE_0:def_10; A47: x c= Intersect FF proof let u be set ; :: according to TARSKI:def_3 ::_thesis: ( not u in x or u in Intersect FF ) assume A48: u in x ; ::_thesis: u in Intersect FF for y being set st y in FF holds u in y proof let y be set ; ::_thesis: ( y in FF implies u in y ) assume A49: y in FF ; ::_thesis: u in y now__::_thesis:_(_(_y_=_(((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_._D_&_u_in_y_)_or_(_y_=_(((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_._B_&_u_in_y_)_or_(_y_=_(((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_._C_&_u_in_y_)_or_(_y_=_(((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_._E_&_u_in_y_)_or_(_y_=_(((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_._F_&_u_in_y_)_) percases ( y = (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . D or y = (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . B or y = (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . C or y = (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . E or y = (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . F ) by A37, A49, ENUMSET1:def_3; caseA50: y = (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . D ; ::_thesis: u in y u in (d /\ ((b /\ c) /\ e)) /\ f by A19, A22, A25, A28, A48, XBOOLE_1:16; then A51: u in d /\ (((b /\ c) /\ e) /\ f) by XBOOLE_1:16; y = d by A6, A50, Th26; hence u in y by A51, XBOOLE_0:def_4; ::_thesis: verum end; caseA52: y = (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . B ; ::_thesis: u in y u in ((c /\ (d /\ b)) /\ e) /\ f by A19, A22, A25, A28, A48, XBOOLE_1:16; then u in (c /\ ((d /\ b) /\ e)) /\ f by XBOOLE_1:16; then u in (c /\ ((d /\ e) /\ b)) /\ f by XBOOLE_1:16; then u in c /\ (((d /\ e) /\ b) /\ f) by XBOOLE_1:16; then u in c /\ ((d /\ e) /\ (f /\ b)) by XBOOLE_1:16; then u in (c /\ (d /\ e)) /\ (f /\ b) by XBOOLE_1:16; then A53: u in ((c /\ (d /\ e)) /\ f) /\ b by XBOOLE_1:16; y = b by A6, A52, Th26; hence u in y by A53, XBOOLE_0:def_4; ::_thesis: verum end; caseA54: y = (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . C ; ::_thesis: u in y u in ((c /\ (b /\ d)) /\ e) /\ f by A19, A22, A25, A28, A48, XBOOLE_1:16; then u in (c /\ ((b /\ d) /\ e)) /\ f by XBOOLE_1:16; then A55: u in c /\ (((b /\ d) /\ e) /\ f) by XBOOLE_1:16; y = c by A6, A54, Th26; hence u in y by A55, XBOOLE_0:def_4; ::_thesis: verum end; case y = (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . E ; ::_thesis: u in y then A56: y = e by A6, Th26; u in (((b /\ c) /\ d) /\ f) /\ e by A19, A22, A25, A28, A48, XBOOLE_1:16; hence u in y by A56, XBOOLE_0:def_4; ::_thesis: verum end; case y = (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . F ; ::_thesis: u in y hence u in y by A19, A31, A48, XBOOLE_0:def_4; ::_thesis: verum end; end; end; hence u in y ; ::_thesis: verum end; then u in meet FF by A46, SETFAM_1:def_1; hence u in Intersect FF by A46, SETFAM_1:def_9; ::_thesis: verum end; A57: for p being set st p in G \ {A} holds (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . p in p proof let p be set ; ::_thesis: ( p in G \ {A} implies (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . p in p ) assume A58: p in G \ {A} ; ::_thesis: (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . p in p now__::_thesis:_(_(_p_=_D_&_(((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_._p_in_p_)_or_(_p_=_B_&_(((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_._p_in_p_)_or_(_p_=_C_&_(((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_._p_in_p_)_or_(_p_=_E_&_(((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_._p_in_p_)_or_(_p_=_F_&_(((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_._p_in_p_)_) percases ( p = D or p = B or p = C or p = E or p = F ) by A7, A12, A58, ENUMSET1:def_3; case p = D ; ::_thesis: (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . p in p hence (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . p in p by A6, A24, Th26; ::_thesis: verum end; case p = B ; ::_thesis: (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . p in p hence (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . p in p by A6, A26, Th26; ::_thesis: verum end; case p = C ; ::_thesis: (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . p in p hence (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . p in p by A6, A27, Th26; ::_thesis: verum end; case p = E ; ::_thesis: (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . p in p hence (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . p in p by A6, A21, Th26; ::_thesis: verum end; case p = F ; ::_thesis: (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . p in p hence (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . p in p by A6, A18, Th26; ::_thesis: verum end; end; end; hence (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . p in p ; ::_thesis: verum end; Intersect FF c= x proof let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in Intersect FF or t in x ) assume A59: t in Intersect FF ; ::_thesis: t in x (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . C in rng (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) by A46, ENUMSET1:def_3; then A60: t in c by A36, A43, A59, SETFAM_1:def_1; (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . B in rng (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) by A46, ENUMSET1:def_3; then t in b by A29, A43, A59, SETFAM_1:def_1; then A61: t in b /\ c by A60, XBOOLE_0:def_4; (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . D in rng (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) by A46, ENUMSET1:def_3; then t in d by A40, A43, A59, SETFAM_1:def_1; then A62: t in (b /\ c) /\ d by A61, XBOOLE_0:def_4; (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . E in rng (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) by A46, ENUMSET1:def_3; then t in e by A30, A43, A59, SETFAM_1:def_1; then A63: t in ((b /\ c) /\ d) /\ e by A62, XBOOLE_0:def_4; (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) . F in rng (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) by A46, ENUMSET1:def_3; then t in f by A31, A43, A59, SETFAM_1:def_1; hence t in x by A19, A22, A25, A28, A63, XBOOLE_0:def_4; ::_thesis: verum end; then x = Intersect FF by A47, XBOOLE_0:def_10; hence x in '/\' (G \ {A}) by A7, A13, A12, A32, A57, A16, BVFUNC_2:def_1; ::_thesis: verum end; A64: '/\' (G \ {A}) c= (((B '/\' C) '/\' D) '/\' E) '/\' F proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in '/\' (G \ {A}) or x in (((B '/\' C) '/\' D) '/\' E) '/\' F ) assume x in '/\' (G \ {A}) ; ::_thesis: x in (((B '/\' C) '/\' D) '/\' E) '/\' F then consider h being Function, FF being Subset-Family of Y such that A65: dom h = G \ {A} and A66: rng h = FF and A67: for d being set st d in G \ {A} holds h . d in d and A68: x = Intersect FF and A69: x <> {} by BVFUNC_2:def_1; A70: C in G \ {A} by A7, A13, A12, ENUMSET1:def_3; then A71: h . C in C by A67; set mbc = (h . B) /\ (h . C); A72: B in G \ {A} by A7, A13, A12, ENUMSET1:def_3; then h . B in B by A67; then A73: (h . B) /\ (h . C) in INTERSECTION (B,C) by A71, SETFAM_1:def_5; set mbcd = ((h . B) /\ (h . C)) /\ (h . D); A74: E in G \ {A} by A7, A13, A12, ENUMSET1:def_3; then A75: h . E in rng h by A65, FUNCT_1:def_3; A76: h . B in rng h by A65, A72, FUNCT_1:def_3; then A77: Intersect FF = meet (rng h) by A66, SETFAM_1:def_9; A78: h . C in rng h by A65, A70, FUNCT_1:def_3; A79: F in G \ {A} by A7, A13, A12, ENUMSET1:def_3; then A80: h . F in rng h by A65, FUNCT_1:def_3; A81: D in G \ {A} by A7, A13, A12, ENUMSET1:def_3; then A82: h . D in rng h by A65, FUNCT_1:def_3; A83: x c= ((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F) proof let m be set ; :: according to TARSKI:def_3 ::_thesis: ( not m in x or m in ((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F) ) assume A84: m in x ; ::_thesis: m in ((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F) then ( m in h . B & m in h . C ) by A68, A76, A78, A77, SETFAM_1:def_1; then A85: m in (h . B) /\ (h . C) by XBOOLE_0:def_4; m in h . D by A68, A82, A77, A84, SETFAM_1:def_1; then A86: m in ((h . B) /\ (h . C)) /\ (h . D) by A85, XBOOLE_0:def_4; m in h . E by A68, A75, A77, A84, SETFAM_1:def_1; then A87: m in (((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E) by A86, XBOOLE_0:def_4; m in h . F by A68, A80, A77, A84, SETFAM_1:def_1; hence m in ((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F) by A87, XBOOLE_0:def_4; ::_thesis: verum end; then ((h . B) /\ (h . C)) /\ (h . D) <> {} by A69; then A88: not ((h . B) /\ (h . C)) /\ (h . D) in {{}} by TARSKI:def_1; A89: rng h <> {} by A65, A72, FUNCT_1:3; ((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F) c= x proof let m be set ; :: according to TARSKI:def_3 ::_thesis: ( not m in ((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F) or m in x ) assume A90: m in ((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F) ; ::_thesis: m in x then A91: m in (((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E) by XBOOLE_0:def_4; then A92: m in ((h . B) /\ (h . C)) /\ (h . D) by XBOOLE_0:def_4; then A93: m in (h . B) /\ (h . C) by XBOOLE_0:def_4; A94: rng h c= {(h . B),(h . C),(h . D),(h . E),(h . F)} proof let u be set ; :: according to TARSKI:def_3 ::_thesis: ( not u in rng h or u in {(h . B),(h . C),(h . D),(h . E),(h . F)} ) assume u in rng h ; ::_thesis: u in {(h . B),(h . C),(h . D),(h . E),(h . F)} then consider x1 being set such that A95: x1 in dom h and A96: u = h . x1 by FUNCT_1:def_3; now__::_thesis:_(_(_x1_=_B_&_u_in_{(h_._B),(h_._C),(h_._D),(h_._E),(h_._F)}_)_or_(_x1_=_C_&_u_in_{(h_._B),(h_._C),(h_._D),(h_._E),(h_._F)}_)_or_(_x1_=_D_&_u_in_{(h_._B),(h_._C),(h_._D),(h_._E),(h_._F)}_)_or_(_x1_=_E_&_u_in_{(h_._B),(h_._C),(h_._D),(h_._E),(h_._F)}_)_or_(_x1_=_F_&_u_in_{(h_._B),(h_._C),(h_._D),(h_._E),(h_._F)}_)_) percases ( x1 = B or x1 = C or x1 = D or x1 = E or x1 = F ) by A7, A12, A65, A95, ENUMSET1:def_3; case x1 = B ; ::_thesis: u in {(h . B),(h . C),(h . D),(h . E),(h . F)} hence u in {(h . B),(h . C),(h . D),(h . E),(h . F)} by A96, ENUMSET1:def_3; ::_thesis: verum end; case x1 = C ; ::_thesis: u in {(h . B),(h . C),(h . D),(h . E),(h . F)} hence u in {(h . B),(h . C),(h . D),(h . E),(h . F)} by A96, ENUMSET1:def_3; ::_thesis: verum end; case x1 = D ; ::_thesis: u in {(h . B),(h . C),(h . D),(h . E),(h . F)} hence u in {(h . B),(h . C),(h . D),(h . E),(h . F)} by A96, ENUMSET1:def_3; ::_thesis: verum end; case x1 = E ; ::_thesis: u in {(h . B),(h . C),(h . D),(h . E),(h . F)} hence u in {(h . B),(h . C),(h . D),(h . E),(h . F)} by A96, ENUMSET1:def_3; ::_thesis: verum end; case x1 = F ; ::_thesis: u in {(h . B),(h . C),(h . D),(h . E),(h . F)} hence u in {(h . B),(h . C),(h . D),(h . E),(h . F)} by A96, ENUMSET1:def_3; ::_thesis: verum end; end; end; hence u in {(h . B),(h . C),(h . D),(h . E),(h . F)} ; ::_thesis: verum end; for y being set st y in rng h holds m in y proof let y be set ; ::_thesis: ( y in rng h implies m in y ) assume A97: y in rng h ; ::_thesis: m in y now__::_thesis:_(_(_y_=_h_._B_&_m_in_y_)_or_(_y_=_h_._C_&_m_in_y_)_or_(_y_=_h_._D_&_m_in_y_)_or_(_y_=_h_._E_&_m_in_y_)_or_(_y_=_h_._F_&_m_in_y_)_) percases ( y = h . B or y = h . C or y = h . D or y = h . E or y = h . F ) by A94, A97, ENUMSET1:def_3; case y = h . B ; ::_thesis: m in y hence m in y by A93, XBOOLE_0:def_4; ::_thesis: verum end; case y = h . C ; ::_thesis: m in y hence m in y by A93, XBOOLE_0:def_4; ::_thesis: verum end; case y = h . D ; ::_thesis: m in y hence m in y by A92, XBOOLE_0:def_4; ::_thesis: verum end; case y = h . E ; ::_thesis: m in y hence m in y by A91, XBOOLE_0:def_4; ::_thesis: verum end; case y = h . F ; ::_thesis: m in y hence m in y by A90, XBOOLE_0:def_4; ::_thesis: verum end; end; end; hence m in y ; ::_thesis: verum end; hence m in x by A68, A89, A77, SETFAM_1:def_1; ::_thesis: verum end; then A98: ((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F) = x by A83, XBOOLE_0:def_10; (h . B) /\ (h . C) <> {} by A69, A83; then not (h . B) /\ (h . C) in {{}} by TARSKI:def_1; then (h . B) /\ (h . C) in (INTERSECTION (B,C)) \ {{}} by A73, XBOOLE_0:def_5; then A99: (h . B) /\ (h . C) in B '/\' C by PARTIT1:def_4; h . D in D by A67, A81; then ((h . B) /\ (h . C)) /\ (h . D) in INTERSECTION ((B '/\' C),D) by A99, SETFAM_1:def_5; then ((h . B) /\ (h . C)) /\ (h . D) in (INTERSECTION ((B '/\' C),D)) \ {{}} by A88, XBOOLE_0:def_5; then A100: ((h . B) /\ (h . C)) /\ (h . D) in (B '/\' C) '/\' D by PARTIT1:def_4; set mbcde = (((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E); A101: not x in {{}} by A69, TARSKI:def_1; (((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E) <> {} by A69, A83; then A102: not (((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E) in {{}} by TARSKI:def_1; h . E in E by A67, A74; then (((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E) in INTERSECTION (((B '/\' C) '/\' D),E) by A100, SETFAM_1:def_5; then (((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E) in (INTERSECTION (((B '/\' C) '/\' D),E)) \ {{}} by A102, XBOOLE_0:def_5; then A103: (((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E) in ((B '/\' C) '/\' D) '/\' E by PARTIT1:def_4; h . F in F by A67, A79; then x in INTERSECTION ((((B '/\' C) '/\' D) '/\' E),F) by A98, A103, SETFAM_1:def_5; then x in (INTERSECTION ((((B '/\' C) '/\' D) '/\' E),F)) \ {{}} by A101, XBOOLE_0:def_5; hence x in (((B '/\' C) '/\' D) '/\' E) '/\' F by PARTIT1:def_4; ::_thesis: verum end; CompF (A,G) = '/\' (G \ {A}) by BVFUNC_2:def_7; hence CompF (A,G) = (((B '/\' C) '/\' D) '/\' E) '/\' F by A14, A64, XBOOLE_0:def_10; ::_thesis: verum end; theorem Th32: :: BVFUNC14:32 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F being a_partition of Y st G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F holds CompF (B,G) = (((A '/\' C) '/\' D) '/\' E) '/\' F proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F being a_partition of Y st G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F holds CompF (B,G) = (((A '/\' C) '/\' D) '/\' E) '/\' F let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C, D, E, F being a_partition of Y st G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F holds CompF (B,G) = (((A '/\' C) '/\' D) '/\' E) '/\' F let A, B, C, D, E, F be a_partition of Y; ::_thesis: ( G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F implies CompF (B,G) = (((A '/\' C) '/\' D) '/\' E) '/\' F ) assume that A1: G = {A,B,C,D,E,F} and A2: ( A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F ) ; ::_thesis: CompF (B,G) = (((A '/\' C) '/\' D) '/\' E) '/\' F {A,B,C,D,E,F} = {B,A} \/ {C,D,E,F} by ENUMSET1:12; then G = {B,A,C,D,E,F} by A1, ENUMSET1:12; hence CompF (B,G) = (((A '/\' C) '/\' D) '/\' E) '/\' F by A2, Th31; ::_thesis: verum end; theorem Th33: :: BVFUNC14:33 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F being a_partition of Y st G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F holds CompF (C,G) = (((A '/\' B) '/\' D) '/\' E) '/\' F proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F being a_partition of Y st G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F holds CompF (C,G) = (((A '/\' B) '/\' D) '/\' E) '/\' F let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C, D, E, F being a_partition of Y st G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F holds CompF (C,G) = (((A '/\' B) '/\' D) '/\' E) '/\' F let A, B, C, D, E, F be a_partition of Y; ::_thesis: ( G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F implies CompF (C,G) = (((A '/\' B) '/\' D) '/\' E) '/\' F ) A1: {A,B,C,D,E,F} = {A,B,C} \/ {D,E,F} by ENUMSET1:13 .= ({A} \/ {B,C}) \/ {D,E,F} by ENUMSET1:2 .= {A,C,B} \/ {D,E,F} by ENUMSET1:2 .= {A,C,B,D,E,F} by ENUMSET1:13 ; assume ( G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F ) ; ::_thesis: CompF (C,G) = (((A '/\' B) '/\' D) '/\' E) '/\' F hence CompF (C,G) = (((A '/\' B) '/\' D) '/\' E) '/\' F by A1, Th32; ::_thesis: verum end; theorem Th34: :: BVFUNC14:34 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F being a_partition of Y st G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F holds CompF (D,G) = (((A '/\' B) '/\' C) '/\' E) '/\' F proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F being a_partition of Y st G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F holds CompF (D,G) = (((A '/\' B) '/\' C) '/\' E) '/\' F let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C, D, E, F being a_partition of Y st G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F holds CompF (D,G) = (((A '/\' B) '/\' C) '/\' E) '/\' F let A, B, C, D, E, F be a_partition of Y; ::_thesis: ( G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F implies CompF (D,G) = (((A '/\' B) '/\' C) '/\' E) '/\' F ) A1: {A,B,C,D,E,F} = {A,B} \/ {C,D,E,F} by ENUMSET1:12 .= {A,B} \/ ({C,D} \/ {E,F}) by ENUMSET1:5 .= {A,B} \/ {D,C,E,F} by ENUMSET1:5 .= {A,B,D,C,E,F} by ENUMSET1:12 ; assume ( G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F ) ; ::_thesis: CompF (D,G) = (((A '/\' B) '/\' C) '/\' E) '/\' F hence CompF (D,G) = (((A '/\' B) '/\' C) '/\' E) '/\' F by A1, Th33; ::_thesis: verum end; theorem Th35: :: BVFUNC14:35 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F being a_partition of Y st G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F holds CompF (E,G) = (((A '/\' B) '/\' C) '/\' D) '/\' F proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F being a_partition of Y st G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F holds CompF (E,G) = (((A '/\' B) '/\' C) '/\' D) '/\' F let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C, D, E, F being a_partition of Y st G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F holds CompF (E,G) = (((A '/\' B) '/\' C) '/\' D) '/\' F let A, B, C, D, E, F be a_partition of Y; ::_thesis: ( G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F implies CompF (E,G) = (((A '/\' B) '/\' C) '/\' D) '/\' F ) A1: {A,B,C,D,E,F} = {A,B,C} \/ {D,E,F} by ENUMSET1:13 .= {A,B,C} \/ ({D,E} \/ {F}) by ENUMSET1:3 .= {A,B,C} \/ {E,D,F} by ENUMSET1:3 .= {A,B,C,E,D,F} by ENUMSET1:13 ; assume ( G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F ) ; ::_thesis: CompF (E,G) = (((A '/\' B) '/\' C) '/\' D) '/\' F hence CompF (E,G) = (((A '/\' B) '/\' C) '/\' D) '/\' F by A1, Th34; ::_thesis: verum end; theorem :: BVFUNC14:36 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F being a_partition of Y st G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F holds CompF (F,G) = (((A '/\' B) '/\' C) '/\' D) '/\' E proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F being a_partition of Y st G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F holds CompF (F,G) = (((A '/\' B) '/\' C) '/\' D) '/\' E let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C, D, E, F being a_partition of Y st G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F holds CompF (F,G) = (((A '/\' B) '/\' C) '/\' D) '/\' E let A, B, C, D, E, F be a_partition of Y; ::_thesis: ( G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F implies CompF (F,G) = (((A '/\' B) '/\' C) '/\' D) '/\' E ) A1: {A,B,C,D,E,F} = {A,B,C,D} \/ {E,F} by ENUMSET1:14 .= {A,B,C,D,F,E} by ENUMSET1:14 ; assume ( G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F ) ; ::_thesis: CompF (F,G) = (((A '/\' B) '/\' C) '/\' D) '/\' E hence CompF (F,G) = (((A '/\' B) '/\' C) '/\' D) '/\' E by A1, Th35; ::_thesis: verum end; theorem Th37: :: BVFUNC14:37 for A, B, C, D, E, F being set for h being Function for A9, B9, C9, D9, E9, F9 being set st A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F & h = (((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (A .--> A9) holds ( h . A = A9 & h . B = B9 & h . C = C9 & h . D = D9 & h . E = E9 & h . F = F9 ) proof let A, B, C, D, E, F be set ; ::_thesis: for h being Function for A9, B9, C9, D9, E9, F9 being set st A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F & h = (((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (A .--> A9) holds ( h . A = A9 & h . B = B9 & h . C = C9 & h . D = D9 & h . E = E9 & h . F = F9 ) let h be Function; ::_thesis: for A9, B9, C9, D9, E9, F9 being set st A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F & h = (((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (A .--> A9) holds ( h . A = A9 & h . B = B9 & h . C = C9 & h . D = D9 & h . E = E9 & h . F = F9 ) let A9, B9, C9, D9, E9, F9 be set ; ::_thesis: ( A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F & h = (((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (A .--> A9) implies ( h . A = A9 & h . B = B9 & h . C = C9 & h . D = D9 & h . E = E9 & h . F = F9 ) ) assume that A1: A <> B and A2: A <> C and A3: A <> D and A4: A <> E and A5: A <> F and A6: ( B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F ) and A7: h = (((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (A .--> A9) ; ::_thesis: ( h . A = A9 & h . B = B9 & h . C = C9 & h . D = D9 & h . E = E9 & h . F = F9 ) A8: dom (A .--> A9) = {A} by FUNCOP_1:13; then A in dom (A .--> A9) by TARSKI:def_1; then A9: h . A = (A .--> A9) . A by A7, FUNCT_4:13; not C in dom (A .--> A9) by A2, A8, TARSKI:def_1; then A10: h . C = (((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) . C by A7, FUNCT_4:11; not F in dom (A .--> A9) by A5, A8, TARSKI:def_1; then A11: h . F = (((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) . F by A7, FUNCT_4:11 .= F9 by A6, Th26 ; not E in dom (A .--> A9) by A4, A8, TARSKI:def_1; then A12: h . E = (((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) . E by A7, FUNCT_4:11 .= E9 by A6, Th26 ; not D in dom (A .--> A9) by A3, A8, TARSKI:def_1; then A13: h . D = (((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) . D by A7, FUNCT_4:11 .= D9 by A6, Th26 ; not B in dom (A .--> A9) by A1, A8, TARSKI:def_1; then h . B = (((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) . B by A7, FUNCT_4:11 .= B9 by A6, Th26 ; hence ( h . A = A9 & h . B = B9 & h . C = C9 & h . D = D9 & h . E = E9 & h . F = F9 ) by A6, A9, A10, A13, A12, A11, Th26, FUNCOP_1:72; ::_thesis: verum end; theorem Th38: :: BVFUNC14:38 for A, B, C, D, E, F being set for h being Function for A9, B9, C9, D9, E9, F9 being set st h = (((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (A .--> A9) holds dom h = {A,B,C,D,E,F} proof let A, B, C, D, E, F be set ; ::_thesis: for h being Function for A9, B9, C9, D9, E9, F9 being set st h = (((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (A .--> A9) holds dom h = {A,B,C,D,E,F} let h be Function; ::_thesis: for A9, B9, C9, D9, E9, F9 being set st h = (((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (A .--> A9) holds dom h = {A,B,C,D,E,F} let A9, B9, C9, D9, E9, F9 be set ; ::_thesis: ( h = (((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (A .--> A9) implies dom h = {A,B,C,D,E,F} ) assume A1: h = (((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (A .--> A9) ; ::_thesis: dom h = {A,B,C,D,E,F} A2: dom (A .--> A9) = {A} by FUNCOP_1:13; dom (((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) = {F,B,C,D,E} by Th27 .= {F} \/ {B,C,D,E} by ENUMSET1:7 .= {B,C,D,E,F} by ENUMSET1:10 ; then dom ((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (A .--> A9)) = {B,C,D,E,F} \/ {A} by A2, FUNCT_4:def_1 .= {A,B,C,D,E,F} by ENUMSET1:11 ; hence dom h = {A,B,C,D,E,F} by A1; ::_thesis: verum end; theorem Th39: :: BVFUNC14:39 for A, B, C, D, E, F being set for h being Function for A9, B9, C9, D9, E9, F9 being set st h = (((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (A .--> A9) holds rng h = {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F)} proof let A, B, C, D, E, F be set ; ::_thesis: for h being Function for A9, B9, C9, D9, E9, F9 being set st h = (((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (A .--> A9) holds rng h = {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F)} let h be Function; ::_thesis: for A9, B9, C9, D9, E9, F9 being set st h = (((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (A .--> A9) holds rng h = {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F)} let A9, B9, C9, D9, E9, F9 be set ; ::_thesis: ( h = (((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (A .--> A9) implies rng h = {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F)} ) assume h = (((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (A .--> A9) ; ::_thesis: rng h = {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F)} then A1: dom h = {A,B,C,D,E,F} by Th38; then A2: B in dom h by ENUMSET1:def_4; A3: rng h c= {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F)} proof let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in rng h or t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F)} ) assume t in rng h ; ::_thesis: t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F)} then consider x1 being set such that A4: x1 in dom h and A5: t = h . x1 by FUNCT_1:def_3; now__::_thesis:_(_(_x1_=_A_&_t_in_{(h_._A),(h_._B),(h_._C),(h_._D),(h_._E),(h_._F)}_)_or_(_x1_=_B_&_t_in_{(h_._A),(h_._B),(h_._C),(h_._D),(h_._E),(h_._F)}_)_or_(_x1_=_C_&_t_in_{(h_._A),(h_._B),(h_._C),(h_._D),(h_._E),(h_._F)}_)_or_(_x1_=_D_&_t_in_{(h_._A),(h_._B),(h_._C),(h_._D),(h_._E),(h_._F)}_)_or_(_x1_=_E_&_t_in_{(h_._A),(h_._B),(h_._C),(h_._D),(h_._E),(h_._F)}_)_or_(_x1_=_F_&_t_in_{(h_._A),(h_._B),(h_._C),(h_._D),(h_._E),(h_._F)}_)_) percases ( x1 = A or x1 = B or x1 = C or x1 = D or x1 = E or x1 = F ) by A1, A4, ENUMSET1:def_4; case x1 = A ; ::_thesis: t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F)} hence t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F)} by A5, ENUMSET1:def_4; ::_thesis: verum end; case x1 = B ; ::_thesis: t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F)} hence t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F)} by A5, ENUMSET1:def_4; ::_thesis: verum end; case x1 = C ; ::_thesis: t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F)} hence t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F)} by A5, ENUMSET1:def_4; ::_thesis: verum end; case x1 = D ; ::_thesis: t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F)} hence t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F)} by A5, ENUMSET1:def_4; ::_thesis: verum end; case x1 = E ; ::_thesis: t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F)} hence t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F)} by A5, ENUMSET1:def_4; ::_thesis: verum end; case x1 = F ; ::_thesis: t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F)} hence t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F)} by A5, ENUMSET1:def_4; ::_thesis: verum end; end; end; hence t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F)} ; ::_thesis: verum end; A6: D in dom h by A1, ENUMSET1:def_4; A7: C in dom h by A1, ENUMSET1:def_4; A8: F in dom h by A1, ENUMSET1:def_4; A9: E in dom h by A1, ENUMSET1:def_4; A10: A in dom h by A1, ENUMSET1:def_4; {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F)} c= rng h proof let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F)} or t in rng h ) assume A11: t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F)} ; ::_thesis: t in rng h now__::_thesis:_(_(_t_=_h_._A_&_t_in_rng_h_)_or_(_t_=_h_._B_&_t_in_rng_h_)_or_(_t_=_h_._C_&_t_in_rng_h_)_or_(_t_=_h_._D_&_t_in_rng_h_)_or_(_t_=_h_._E_&_t_in_rng_h_)_or_(_t_=_h_._F_&_t_in_rng_h_)_) percases ( t = h . A or t = h . B or t = h . C or t = h . D or t = h . E or t = h . F ) by A11, ENUMSET1:def_4; case t = h . A ; ::_thesis: t in rng h hence t in rng h by A10, FUNCT_1:def_3; ::_thesis: verum end; case t = h . B ; ::_thesis: t in rng h hence t in rng h by A2, FUNCT_1:def_3; ::_thesis: verum end; case t = h . C ; ::_thesis: t in rng h hence t in rng h by A7, FUNCT_1:def_3; ::_thesis: verum end; case t = h . D ; ::_thesis: t in rng h hence t in rng h by A6, FUNCT_1:def_3; ::_thesis: verum end; case t = h . E ; ::_thesis: t in rng h hence t in rng h by A9, FUNCT_1:def_3; ::_thesis: verum end; case t = h . F ; ::_thesis: t in rng h hence t in rng h by A8, FUNCT_1:def_3; ::_thesis: verum end; end; end; hence t in rng h ; ::_thesis: verum end; hence rng h = {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F)} by A3, XBOOLE_0:def_10; ::_thesis: verum end; theorem :: BVFUNC14:40 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F being a_partition of Y for z, u being Element of Y for h being Function st G is independent & G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F holds EqClass (u,((((B '/\' C) '/\' D) '/\' E) '/\' F)) meets EqClass (z,A) proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F being a_partition of Y for z, u being Element of Y for h being Function st G is independent & G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F holds EqClass (u,((((B '/\' C) '/\' D) '/\' E) '/\' F)) meets EqClass (z,A) let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C, D, E, F being a_partition of Y for z, u being Element of Y for h being Function st G is independent & G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F holds EqClass (u,((((B '/\' C) '/\' D) '/\' E) '/\' F)) meets EqClass (z,A) let A, B, C, D, E, F be a_partition of Y; ::_thesis: for z, u being Element of Y for h being Function st G is independent & G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F holds EqClass (u,((((B '/\' C) '/\' D) '/\' E) '/\' F)) meets EqClass (z,A) let z, u be Element of Y; ::_thesis: for h being Function st G is independent & G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F holds EqClass (u,((((B '/\' C) '/\' D) '/\' E) '/\' F)) meets EqClass (z,A) let h be Function; ::_thesis: ( G is independent & G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F implies EqClass (u,((((B '/\' C) '/\' D) '/\' E) '/\' F)) meets EqClass (z,A) ) assume that A1: G is independent and A2: G = {A,B,C,D,E,F} and A3: ( A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F ) ; ::_thesis: EqClass (u,((((B '/\' C) '/\' D) '/\' E) '/\' F)) meets EqClass (z,A) set h = (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A))); A4: ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . A = EqClass (z,A) by A3, Th37; set GG = EqClass (u,((((B '/\' C) '/\' D) '/\' E) '/\' F)); EqClass (u,((((B '/\' C) '/\' D) '/\' E) '/\' F)) = (EqClass (u,(((B '/\' C) '/\' D) '/\' E))) /\ (EqClass (u,F)) by Th1; then EqClass (u,((((B '/\' C) '/\' D) '/\' E) '/\' F)) = ((EqClass (u,((B '/\' C) '/\' D))) /\ (EqClass (u,E))) /\ (EqClass (u,F)) by Th1; then EqClass (u,((((B '/\' C) '/\' D) '/\' E) '/\' F)) = (((EqClass (u,(B '/\' C))) /\ (EqClass (u,D))) /\ (EqClass (u,E))) /\ (EqClass (u,F)) by Th1; then A5: (EqClass (u,((((B '/\' C) '/\' D) '/\' E) '/\' F))) /\ (EqClass (z,A)) = (((((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E))) /\ (EqClass (u,F))) /\ (EqClass (z,A)) by Th1; A6: ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . B = EqClass (u,B) by A3, Th37; A7: ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . D = EqClass (u,D) by A3, Th37; A8: ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . C = EqClass (u,C) by A3, Th37; A9: ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . F = EqClass (u,F) by A3, Th37; A10: ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . E = EqClass (u,E) by A3, Th37; A11: rng ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) = {(((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . A),(((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . B),(((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . C),(((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . D),(((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . E),(((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . F)} by Th39; rng ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) c= bool Y proof let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in rng ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) or t in bool Y ) assume A12: t in rng ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) ; ::_thesis: t in bool Y now__::_thesis:_(_(_t_=_((((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(D_.-->_(EqClass_(u,D))))_+*_(E_.-->_(EqClass_(u,E))))_+*_(F_.-->_(EqClass_(u,F))))_+*_(A_.-->_(EqClass_(z,A))))_._A_&_t_in_bool_Y_)_or_(_t_=_((((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(D_.-->_(EqClass_(u,D))))_+*_(E_.-->_(EqClass_(u,E))))_+*_(F_.-->_(EqClass_(u,F))))_+*_(A_.-->_(EqClass_(z,A))))_._B_&_t_in_bool_Y_)_or_(_t_=_((((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(D_.-->_(EqClass_(u,D))))_+*_(E_.-->_(EqClass_(u,E))))_+*_(F_.-->_(EqClass_(u,F))))_+*_(A_.-->_(EqClass_(z,A))))_._C_&_t_in_bool_Y_)_or_(_t_=_((((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(D_.-->_(EqClass_(u,D))))_+*_(E_.-->_(EqClass_(u,E))))_+*_(F_.-->_(EqClass_(u,F))))_+*_(A_.-->_(EqClass_(z,A))))_._D_&_t_in_bool_Y_)_or_(_t_=_((((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(D_.-->_(EqClass_(u,D))))_+*_(E_.-->_(EqClass_(u,E))))_+*_(F_.-->_(EqClass_(u,F))))_+*_(A_.-->_(EqClass_(z,A))))_._E_&_t_in_bool_Y_)_or_(_t_=_((((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(D_.-->_(EqClass_(u,D))))_+*_(E_.-->_(EqClass_(u,E))))_+*_(F_.-->_(EqClass_(u,F))))_+*_(A_.-->_(EqClass_(z,A))))_._F_&_t_in_bool_Y_)_) percases ( t = ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . A or t = ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . B or t = ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . C or t = ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . D or t = ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . E or t = ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . F ) by A11, A12, ENUMSET1:def_4; case t = ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . A ; ::_thesis: t in bool Y hence t in bool Y by A4; ::_thesis: verum end; case t = ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . B ; ::_thesis: t in bool Y hence t in bool Y by A6; ::_thesis: verum end; case t = ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . C ; ::_thesis: t in bool Y hence t in bool Y by A8; ::_thesis: verum end; case t = ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . D ; ::_thesis: t in bool Y hence t in bool Y by A7; ::_thesis: verum end; case t = ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . E ; ::_thesis: t in bool Y hence t in bool Y by A10; ::_thesis: verum end; case t = ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . F ; ::_thesis: t in bool Y hence t in bool Y by A9; ::_thesis: verum end; end; end; hence t in bool Y ; ::_thesis: verum end; then reconsider FF = rng ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) as Subset-Family of Y ; A13: dom ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) = G by A2, Th38; then A in dom ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) by A2, ENUMSET1:def_4; then A14: ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . A in rng ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then A15: Intersect FF = meet (rng ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A))))) by SETFAM_1:def_9; for d being set st d in G holds ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . d in d proof let d be set ; ::_thesis: ( d in G implies ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . d in d ) assume A16: d in G ; ::_thesis: ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . d in d now__::_thesis:_(_(_d_=_A_&_((((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(D_.-->_(EqClass_(u,D))))_+*_(E_.-->_(EqClass_(u,E))))_+*_(F_.-->_(EqClass_(u,F))))_+*_(A_.-->_(EqClass_(z,A))))_._d_in_d_)_or_(_d_=_B_&_((((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(D_.-->_(EqClass_(u,D))))_+*_(E_.-->_(EqClass_(u,E))))_+*_(F_.-->_(EqClass_(u,F))))_+*_(A_.-->_(EqClass_(z,A))))_._d_in_d_)_or_(_d_=_C_&_((((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(D_.-->_(EqClass_(u,D))))_+*_(E_.-->_(EqClass_(u,E))))_+*_(F_.-->_(EqClass_(u,F))))_+*_(A_.-->_(EqClass_(z,A))))_._d_in_d_)_or_(_d_=_D_&_((((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(D_.-->_(EqClass_(u,D))))_+*_(E_.-->_(EqClass_(u,E))))_+*_(F_.-->_(EqClass_(u,F))))_+*_(A_.-->_(EqClass_(z,A))))_._d_in_d_)_or_(_d_=_E_&_((((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(D_.-->_(EqClass_(u,D))))_+*_(E_.-->_(EqClass_(u,E))))_+*_(F_.-->_(EqClass_(u,F))))_+*_(A_.-->_(EqClass_(z,A))))_._d_in_d_)_or_(_d_=_F_&_((((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(D_.-->_(EqClass_(u,D))))_+*_(E_.-->_(EqClass_(u,E))))_+*_(F_.-->_(EqClass_(u,F))))_+*_(A_.-->_(EqClass_(z,A))))_._d_in_d_)_) percases ( d = A or d = B or d = C or d = D or d = E or d = F ) by A2, A16, ENUMSET1:def_4; case d = A ; ::_thesis: ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . d in d hence ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . d in d by A4; ::_thesis: verum end; case d = B ; ::_thesis: ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . d in d hence ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . d in d by A6; ::_thesis: verum end; case d = C ; ::_thesis: ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . d in d hence ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . d in d by A8; ::_thesis: verum end; case d = D ; ::_thesis: ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . d in d hence ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . d in d by A7; ::_thesis: verum end; case d = E ; ::_thesis: ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . d in d hence ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . d in d by A10; ::_thesis: verum end; case d = F ; ::_thesis: ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . d in d hence ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . d in d by A9; ::_thesis: verum end; end; end; hence ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . d in d ; ::_thesis: verum end; then Intersect FF <> {} by A1, A13, BVFUNC_2:def_5; then consider m being set such that A17: m in Intersect FF by XBOOLE_0:def_1; C in dom ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) by A2, A13, ENUMSET1:def_4; then ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . C in rng ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then A18: m in EqClass (u,C) by A8, A15, A17, SETFAM_1:def_1; B in dom ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) by A2, A13, ENUMSET1:def_4; then ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . B in rng ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then m in EqClass (u,B) by A6, A15, A17, SETFAM_1:def_1; then A19: m in (EqClass (u,B)) /\ (EqClass (u,C)) by A18, XBOOLE_0:def_4; D in dom ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) by A2, A13, ENUMSET1:def_4; then ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . D in rng ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then m in EqClass (u,D) by A7, A15, A17, SETFAM_1:def_1; then A20: m in ((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D)) by A19, XBOOLE_0:def_4; E in dom ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) by A2, A13, ENUMSET1:def_4; then ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . E in rng ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then m in EqClass (u,E) by A10, A15, A17, SETFAM_1:def_1; then A21: m in (((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E)) by A20, XBOOLE_0:def_4; F in dom ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) by A2, A13, ENUMSET1:def_4; then ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . F in rng ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then m in EqClass (u,F) by A9, A15, A17, SETFAM_1:def_1; then A22: m in ((((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E))) /\ (EqClass (u,F)) by A21, XBOOLE_0:def_4; m in EqClass (z,A) by A4, A14, A15, A17, SETFAM_1:def_1; then m in (((((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E))) /\ (EqClass (u,F))) /\ (EqClass (z,A)) by A22, XBOOLE_0:def_4; hence EqClass (u,((((B '/\' C) '/\' D) '/\' E) '/\' F)) meets EqClass (z,A) by A5, XBOOLE_0:def_7; ::_thesis: verum end; theorem :: BVFUNC14:41 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F being a_partition of Y for z, u being Element of Y for h being Function st G is independent & G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F & EqClass (z,(((C '/\' D) '/\' E) '/\' F)) = EqClass (u,(((C '/\' D) '/\' E) '/\' F)) holds EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G))) proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F being a_partition of Y for z, u being Element of Y for h being Function st G is independent & G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F & EqClass (z,(((C '/\' D) '/\' E) '/\' F)) = EqClass (u,(((C '/\' D) '/\' E) '/\' F)) holds EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G))) let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C, D, E, F being a_partition of Y for z, u being Element of Y for h being Function st G is independent & G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F & EqClass (z,(((C '/\' D) '/\' E) '/\' F)) = EqClass (u,(((C '/\' D) '/\' E) '/\' F)) holds EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G))) let A, B, C, D, E, F be a_partition of Y; ::_thesis: for z, u being Element of Y for h being Function st G is independent & G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F & EqClass (z,(((C '/\' D) '/\' E) '/\' F)) = EqClass (u,(((C '/\' D) '/\' E) '/\' F)) holds EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G))) let z, u be Element of Y; ::_thesis: for h being Function st G is independent & G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F & EqClass (z,(((C '/\' D) '/\' E) '/\' F)) = EqClass (u,(((C '/\' D) '/\' E) '/\' F)) holds EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G))) let h be Function; ::_thesis: ( G is independent & G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F & EqClass (z,(((C '/\' D) '/\' E) '/\' F)) = EqClass (u,(((C '/\' D) '/\' E) '/\' F)) implies EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G))) ) assume that A1: G is independent and A2: G = {A,B,C,D,E,F} and A3: ( A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F ) and A4: EqClass (z,(((C '/\' D) '/\' E) '/\' F)) = EqClass (u,(((C '/\' D) '/\' E) '/\' F)) ; ::_thesis: EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G))) set h = (((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A))); A5: ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . A = EqClass (z,A) by A3, Th37; set I = EqClass (z,A); set GG = EqClass (u,((((B '/\' C) '/\' D) '/\' E) '/\' F)); set H = EqClass (z,(CompF (B,G))); A6: A '/\' (((C '/\' D) '/\' E) '/\' F) = (A '/\' ((C '/\' D) '/\' E)) '/\' F by PARTIT1:14 .= ((A '/\' (C '/\' D)) '/\' E) '/\' F by PARTIT1:14 .= (((A '/\' C) '/\' D) '/\' E) '/\' F by PARTIT1:14 ; EqClass (u,((((B '/\' C) '/\' D) '/\' E) '/\' F)) = (EqClass (u,(((B '/\' C) '/\' D) '/\' E))) /\ (EqClass (u,F)) by Th1; then EqClass (u,((((B '/\' C) '/\' D) '/\' E) '/\' F)) = ((EqClass (u,((B '/\' C) '/\' D))) /\ (EqClass (u,E))) /\ (EqClass (u,F)) by Th1; then EqClass (u,((((B '/\' C) '/\' D) '/\' E) '/\' F)) = (((EqClass (u,(B '/\' C))) /\ (EqClass (u,D))) /\ (EqClass (u,E))) /\ (EqClass (u,F)) by Th1; then A7: (EqClass (u,((((B '/\' C) '/\' D) '/\' E) '/\' F))) /\ (EqClass (z,A)) = (((((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E))) /\ (EqClass (u,F))) /\ (EqClass (z,A)) by Th1; A8: ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . B = EqClass (u,B) by A3, Th37; A9: ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . F = EqClass (u,F) by A3, Th37; A10: ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . E = EqClass (u,E) by A3, Th37; A11: ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . D = EqClass (u,D) by A3, Th37; A12: ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . C = EqClass (u,C) by A3, Th37; A13: rng ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) = {(((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . A),(((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . B),(((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . C),(((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . D),(((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . E),(((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . F)} by Th39; rng ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) c= bool Y proof let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in rng ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) or t in bool Y ) assume A14: t in rng ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) ; ::_thesis: t in bool Y now__::_thesis:_(_(_t_=_((((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(D_.-->_(EqClass_(u,D))))_+*_(E_.-->_(EqClass_(u,E))))_+*_(F_.-->_(EqClass_(u,F))))_+*_(A_.-->_(EqClass_(z,A))))_._A_&_t_in_bool_Y_)_or_(_t_=_((((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(D_.-->_(EqClass_(u,D))))_+*_(E_.-->_(EqClass_(u,E))))_+*_(F_.-->_(EqClass_(u,F))))_+*_(A_.-->_(EqClass_(z,A))))_._B_&_t_in_bool_Y_)_or_(_t_=_((((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(D_.-->_(EqClass_(u,D))))_+*_(E_.-->_(EqClass_(u,E))))_+*_(F_.-->_(EqClass_(u,F))))_+*_(A_.-->_(EqClass_(z,A))))_._C_&_t_in_bool_Y_)_or_(_t_=_((((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(D_.-->_(EqClass_(u,D))))_+*_(E_.-->_(EqClass_(u,E))))_+*_(F_.-->_(EqClass_(u,F))))_+*_(A_.-->_(EqClass_(z,A))))_._D_&_t_in_bool_Y_)_or_(_t_=_((((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(D_.-->_(EqClass_(u,D))))_+*_(E_.-->_(EqClass_(u,E))))_+*_(F_.-->_(EqClass_(u,F))))_+*_(A_.-->_(EqClass_(z,A))))_._E_&_t_in_bool_Y_)_or_(_t_=_((((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(D_.-->_(EqClass_(u,D))))_+*_(E_.-->_(EqClass_(u,E))))_+*_(F_.-->_(EqClass_(u,F))))_+*_(A_.-->_(EqClass_(z,A))))_._F_&_t_in_bool_Y_)_) percases ( t = ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . A or t = ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . B or t = ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . C or t = ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . D or t = ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . E or t = ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . F ) by A13, A14, ENUMSET1:def_4; case t = ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . A ; ::_thesis: t in bool Y hence t in bool Y by A5; ::_thesis: verum end; case t = ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . B ; ::_thesis: t in bool Y hence t in bool Y by A8; ::_thesis: verum end; case t = ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . C ; ::_thesis: t in bool Y hence t in bool Y by A12; ::_thesis: verum end; case t = ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . D ; ::_thesis: t in bool Y hence t in bool Y by A11; ::_thesis: verum end; case t = ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . E ; ::_thesis: t in bool Y hence t in bool Y by A10; ::_thesis: verum end; case t = ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . F ; ::_thesis: t in bool Y hence t in bool Y by A9; ::_thesis: verum end; end; end; hence t in bool Y ; ::_thesis: verum end; then reconsider FF = rng ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) as Subset-Family of Y ; A15: dom ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) = G by A2, Th38; then A in dom ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) by A2, ENUMSET1:def_4; then A16: ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . A in rng ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then A17: Intersect FF = meet (rng ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A))))) by SETFAM_1:def_9; for d being set st d in G holds ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . d in d proof let d be set ; ::_thesis: ( d in G implies ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . d in d ) assume A18: d in G ; ::_thesis: ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . d in d now__::_thesis:_(_(_d_=_A_&_((((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(D_.-->_(EqClass_(u,D))))_+*_(E_.-->_(EqClass_(u,E))))_+*_(F_.-->_(EqClass_(u,F))))_+*_(A_.-->_(EqClass_(z,A))))_._d_in_d_)_or_(_d_=_B_&_((((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(D_.-->_(EqClass_(u,D))))_+*_(E_.-->_(EqClass_(u,E))))_+*_(F_.-->_(EqClass_(u,F))))_+*_(A_.-->_(EqClass_(z,A))))_._d_in_d_)_or_(_d_=_C_&_((((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(D_.-->_(EqClass_(u,D))))_+*_(E_.-->_(EqClass_(u,E))))_+*_(F_.-->_(EqClass_(u,F))))_+*_(A_.-->_(EqClass_(z,A))))_._d_in_d_)_or_(_d_=_D_&_((((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(D_.-->_(EqClass_(u,D))))_+*_(E_.-->_(EqClass_(u,E))))_+*_(F_.-->_(EqClass_(u,F))))_+*_(A_.-->_(EqClass_(z,A))))_._d_in_d_)_or_(_d_=_E_&_((((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(D_.-->_(EqClass_(u,D))))_+*_(E_.-->_(EqClass_(u,E))))_+*_(F_.-->_(EqClass_(u,F))))_+*_(A_.-->_(EqClass_(z,A))))_._d_in_d_)_or_(_d_=_F_&_((((((B_.-->_(EqClass_(u,B)))_+*_(C_.-->_(EqClass_(u,C))))_+*_(D_.-->_(EqClass_(u,D))))_+*_(E_.-->_(EqClass_(u,E))))_+*_(F_.-->_(EqClass_(u,F))))_+*_(A_.-->_(EqClass_(z,A))))_._d_in_d_)_) percases ( d = A or d = B or d = C or d = D or d = E or d = F ) by A2, A18, ENUMSET1:def_4; case d = A ; ::_thesis: ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . d in d hence ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . d in d by A5; ::_thesis: verum end; case d = B ; ::_thesis: ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . d in d hence ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . d in d by A8; ::_thesis: verum end; case d = C ; ::_thesis: ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . d in d hence ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . d in d by A12; ::_thesis: verum end; case d = D ; ::_thesis: ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . d in d hence ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . d in d by A11; ::_thesis: verum end; case d = E ; ::_thesis: ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . d in d hence ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . d in d by A10; ::_thesis: verum end; case d = F ; ::_thesis: ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . d in d hence ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . d in d by A9; ::_thesis: verum end; end; end; hence ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . d in d ; ::_thesis: verum end; then Intersect FF <> {} by A1, A15, BVFUNC_2:def_5; then consider m being set such that A19: m in Intersect FF by XBOOLE_0:def_1; D in dom ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) by A2, A15, ENUMSET1:def_4; then ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . D in rng ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then m in ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . D by A17, A19, SETFAM_1:def_1; then A20: m in EqClass (u,D) by A3, Th37; C in dom ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) by A2, A15, ENUMSET1:def_4; then ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . C in rng ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then m in ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . C by A17, A19, SETFAM_1:def_1; then A21: m in EqClass (u,C) by A3, Th37; B in dom ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) by A2, A15, ENUMSET1:def_4; then ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . B in rng ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then m in ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . B by A17, A19, SETFAM_1:def_1; then m in EqClass (u,B) by A3, Th37; then m in (EqClass (u,B)) /\ (EqClass (u,C)) by A21, XBOOLE_0:def_4; then A22: m in ((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D)) by A20, XBOOLE_0:def_4; F in dom ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) by A2, A15, ENUMSET1:def_4; then ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . F in rng ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then m in ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . F by A17, A19, SETFAM_1:def_1; then A23: m in EqClass (u,F) by A3, Th37; E in dom ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) by A2, A15, ENUMSET1:def_4; then ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . E in rng ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then m in ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . E by A17, A19, SETFAM_1:def_1; then m in EqClass (u,E) by A3, Th37; then m in (((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E)) by A22, XBOOLE_0:def_4; then A24: m in ((((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E))) /\ (EqClass (u,F)) by A23, XBOOLE_0:def_4; m in ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . A by A16, A17, A19, SETFAM_1:def_1; then m in EqClass (z,A) by A3, Th37; then (EqClass (u,((((B '/\' C) '/\' D) '/\' E) '/\' F))) /\ (EqClass (z,A)) <> {} by A7, A24, XBOOLE_0:def_4; then consider p being set such that A25: p in (EqClass (u,((((B '/\' C) '/\' D) '/\' E) '/\' F))) /\ (EqClass (z,A)) by XBOOLE_0:def_1; reconsider p = p as Element of Y by A25; A26: p in EqClass (u,((((B '/\' C) '/\' D) '/\' E) '/\' F)) by A25, XBOOLE_0:def_4; set L = EqClass (z,(((C '/\' D) '/\' E) '/\' F)); EqClass (u,((((B '/\' C) '/\' D) '/\' E) '/\' F)) = EqClass (u,(((B '/\' (C '/\' D)) '/\' E) '/\' F)) by PARTIT1:14; then EqClass (u,((((B '/\' C) '/\' D) '/\' E) '/\' F)) = EqClass (u,((B '/\' ((C '/\' D) '/\' E)) '/\' F)) by PARTIT1:14; then EqClass (u,((((B '/\' C) '/\' D) '/\' E) '/\' F)) = EqClass (u,(B '/\' (((C '/\' D) '/\' E) '/\' F))) by PARTIT1:14; then A27: EqClass (u,((((B '/\' C) '/\' D) '/\' E) '/\' F)) c= EqClass (z,(((C '/\' D) '/\' E) '/\' F)) by A4, BVFUNC11:3; A28: z in EqClass (z,(CompF (B,G))) by EQREL_1:def_6; set K = EqClass (p,(((C '/\' D) '/\' E) '/\' F)); ( p in EqClass (p,(((C '/\' D) '/\' E) '/\' F)) & p in EqClass (z,A) ) by A25, EQREL_1:def_6, XBOOLE_0:def_4; then A29: p in (EqClass (z,A)) /\ (EqClass (p,(((C '/\' D) '/\' E) '/\' F))) by XBOOLE_0:def_4; then ( (EqClass (z,A)) /\ (EqClass (p,(((C '/\' D) '/\' E) '/\' F))) in INTERSECTION (A,(((C '/\' D) '/\' E) '/\' F)) & not (EqClass (z,A)) /\ (EqClass (p,(((C '/\' D) '/\' E) '/\' F))) in {{}} ) by SETFAM_1:def_5, TARSKI:def_1; then (EqClass (z,A)) /\ (EqClass (p,(((C '/\' D) '/\' E) '/\' F))) in (INTERSECTION (A,(((C '/\' D) '/\' E) '/\' F))) \ {{}} by XBOOLE_0:def_5; then A30: (EqClass (z,A)) /\ (EqClass (p,(((C '/\' D) '/\' E) '/\' F))) in A '/\' (((C '/\' D) '/\' E) '/\' F) by PARTIT1:def_4; p in EqClass (p,(((C '/\' D) '/\' E) '/\' F)) by EQREL_1:def_6; then EqClass (p,(((C '/\' D) '/\' E) '/\' F)) meets EqClass (z,(((C '/\' D) '/\' E) '/\' F)) by A27, A26, XBOOLE_0:3; then EqClass (p,(((C '/\' D) '/\' E) '/\' F)) = EqClass (z,(((C '/\' D) '/\' E) '/\' F)) by EQREL_1:41; then A31: z in EqClass (p,(((C '/\' D) '/\' E) '/\' F)) by EQREL_1:def_6; z in EqClass (z,A) by EQREL_1:def_6; then A32: z in (EqClass (z,A)) /\ (EqClass (p,(((C '/\' D) '/\' E) '/\' F))) by A31, XBOOLE_0:def_4; CompF (B,G) = (((A '/\' C) '/\' D) '/\' E) '/\' F by A2, A3, Th32; then A33: ( (EqClass (z,A)) /\ (EqClass (p,(((C '/\' D) '/\' E) '/\' F))) = EqClass (z,(CompF (B,G))) or (EqClass (z,A)) /\ (EqClass (p,(((C '/\' D) '/\' E) '/\' F))) misses EqClass (z,(CompF (B,G))) ) by A30, A6, EQREL_1:def_4; EqClass (u,((((B '/\' C) '/\' D) '/\' E) '/\' F)) = EqClass (u,(CompF (A,G))) by A2, A3, Th31; hence EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G))) by A29, A26, A32, A28, A33, XBOOLE_0:3; ::_thesis: verum end; begin theorem Th42: :: BVFUNC14:42 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J being a_partition of Y st G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J holds CompF (A,G) = ((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J being a_partition of Y st G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J holds CompF (A,G) = ((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C, D, E, F, J being a_partition of Y st G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J holds CompF (A,G) = ((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J let A, B, C, D, E, F, J be a_partition of Y; ::_thesis: ( G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J implies CompF (A,G) = ((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J ) assume that A1: G = {A,B,C,D,E,F,J} and A2: A <> B and A3: A <> C and A4: ( A <> D & A <> E ) and A5: ( A <> F & A <> J ) and A6: ( B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J ) ; ::_thesis: CompF (A,G) = ((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J A7: G \ {A} = ({A} \/ {B,C,D,E,F,J}) \ {A} by A1, ENUMSET1:16; ( not D in {A} & not E in {A} ) by A4, TARSKI:def_1; then A8: {D,E} \ {A} = {D,E} by ZFMISC_1:63; A9: ( not F in {A} & not J in {A} ) by A5, TARSKI:def_1; A10: not C in {A} by A3, TARSKI:def_1; A11: not B in {A} by A2, TARSKI:def_1; {B,C,D,E,F,J} \ {A} = ({B} \/ {C,D,E,F,J}) \ {A} by ENUMSET1:11 .= ({B} \ {A}) \/ ({C,D,E,F,J} \ {A}) by XBOOLE_1:42 .= {B} \/ ({C,D,E,F,J} \ {A}) by A11, ZFMISC_1:59 .= {B} \/ (({C} \/ {D,E,F,J}) \ {A}) by ENUMSET1:7 .= {B} \/ (({C} \ {A}) \/ ({D,E,F,J} \ {A})) by XBOOLE_1:42 .= {B} \/ (({C} \ {A}) \/ (({D,E} \/ {F,J}) \ {A})) by ENUMSET1:5 .= {B} \/ (({C} \ {A}) \/ (({D,E} \ {A}) \/ ({F,J} \ {A}))) by XBOOLE_1:42 .= {B} \/ (({C} \ {A}) \/ ({D,E} \/ {F,J})) by A9, A8, ZFMISC_1:63 .= {B} \/ ({C} \/ ({D,E} \/ {F,J})) by A10, ZFMISC_1:59 .= {B} \/ ({C} \/ {D,E,F,J}) by ENUMSET1:5 .= {B} \/ {C,D,E,F,J} by ENUMSET1:7 .= {B,C,D,E,F,J} by ENUMSET1:11 ; then A12: G \ {A} = ({A} \ {A}) \/ {B,C,D,E,F,J} by A7, XBOOLE_1:42 .= {} \/ {B,C,D,E,F,J} by XBOOLE_1:37 ; A13: '/\' (G \ {A}) c= ((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in '/\' (G \ {A}) or x in ((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J ) assume x in '/\' (G \ {A}) ; ::_thesis: x in ((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J then consider h being Function, FF being Subset-Family of Y such that A14: dom h = G \ {A} and A15: rng h = FF and A16: for d being set st d in G \ {A} holds h . d in d and A17: x = Intersect FF and A18: x <> {} by BVFUNC_2:def_1; A19: C in G \ {A} by A12, ENUMSET1:def_4; then A20: h . C in C by A16; set mbcd = ((h . B) /\ (h . C)) /\ (h . D); A21: E in G \ {A} by A12, ENUMSET1:def_4; then A22: h . E in rng h by A14, FUNCT_1:def_3; set mbc = (h . B) /\ (h . C); A23: B in G \ {A} by A12, ENUMSET1:def_4; then h . B in B by A16; then A24: (h . B) /\ (h . C) in INTERSECTION (B,C) by A20, SETFAM_1:def_5; A25: h . B in rng h by A14, A23, FUNCT_1:def_3; then A26: Intersect FF = meet (rng h) by A15, SETFAM_1:def_9; A27: h . C in rng h by A14, A19, FUNCT_1:def_3; A28: F in G \ {A} by A12, ENUMSET1:def_4; then A29: h . F in rng h by A14, FUNCT_1:def_3; set mbcdef = ((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F); set mbcde = (((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E); A30: not x in {{}} by A18, TARSKI:def_1; A31: J in G \ {A} by A12, ENUMSET1:def_4; then A32: h . J in rng h by A14, FUNCT_1:def_3; A33: D in G \ {A} by A12, ENUMSET1:def_4; then A34: h . D in rng h by A14, FUNCT_1:def_3; A35: x c= (((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F)) /\ (h . J) proof let m be set ; :: according to TARSKI:def_3 ::_thesis: ( not m in x or m in (((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F)) /\ (h . J) ) assume A36: m in x ; ::_thesis: m in (((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F)) /\ (h . J) then ( m in h . B & m in h . C ) by A17, A25, A27, A26, SETFAM_1:def_1; then A37: m in (h . B) /\ (h . C) by XBOOLE_0:def_4; m in h . D by A17, A34, A26, A36, SETFAM_1:def_1; then A38: m in ((h . B) /\ (h . C)) /\ (h . D) by A37, XBOOLE_0:def_4; m in h . E by A17, A22, A26, A36, SETFAM_1:def_1; then A39: m in (((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E) by A38, XBOOLE_0:def_4; m in h . F by A17, A29, A26, A36, SETFAM_1:def_1; then A40: m in ((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F) by A39, XBOOLE_0:def_4; m in h . J by A17, A32, A26, A36, SETFAM_1:def_1; hence m in (((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F)) /\ (h . J) by A40, XBOOLE_0:def_4; ::_thesis: verum end; then ((h . B) /\ (h . C)) /\ (h . D) <> {} by A18; then A41: not ((h . B) /\ (h . C)) /\ (h . D) in {{}} by TARSKI:def_1; (((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F)) /\ (h . J) c= x proof A42: rng h c= {(h . B),(h . C),(h . D),(h . E),(h . F),(h . J)} proof let u be set ; :: according to TARSKI:def_3 ::_thesis: ( not u in rng h or u in {(h . B),(h . C),(h . D),(h . E),(h . F),(h . J)} ) assume u in rng h ; ::_thesis: u in {(h . B),(h . C),(h . D),(h . E),(h . F),(h . J)} then consider x1 being set such that A43: x1 in dom h and A44: u = h . x1 by FUNCT_1:def_3; ( x1 = B or x1 = C or x1 = D or x1 = E or x1 = F or x1 = J ) by A12, A14, A43, ENUMSET1:def_4; hence u in {(h . B),(h . C),(h . D),(h . E),(h . F),(h . J)} by A44, ENUMSET1:def_4; ::_thesis: verum end; let m be set ; :: according to TARSKI:def_3 ::_thesis: ( not m in (((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F)) /\ (h . J) or m in x ) assume A45: m in (((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F)) /\ (h . J) ; ::_thesis: m in x then A46: m in h . J by XBOOLE_0:def_4; A47: m in ((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F) by A45, XBOOLE_0:def_4; then A48: m in (((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E) by XBOOLE_0:def_4; then A49: m in h . E by XBOOLE_0:def_4; A50: m in ((h . B) /\ (h . C)) /\ (h . D) by A48, XBOOLE_0:def_4; then A51: m in h . D by XBOOLE_0:def_4; m in (h . B) /\ (h . C) by A50, XBOOLE_0:def_4; then A52: ( m in h . B & m in h . C ) by XBOOLE_0:def_4; m in h . F by A47, XBOOLE_0:def_4; then for y being set st y in rng h holds m in y by A52, A51, A49, A46, A42, ENUMSET1:def_4; hence m in x by A17, A25, A26, SETFAM_1:def_1; ::_thesis: verum end; then A53: (((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F)) /\ (h . J) = x by A35, XBOOLE_0:def_10; (h . B) /\ (h . C) <> {} by A18, A35; then not (h . B) /\ (h . C) in {{}} by TARSKI:def_1; then (h . B) /\ (h . C) in (INTERSECTION (B,C)) \ {{}} by A24, XBOOLE_0:def_5; then A54: (h . B) /\ (h . C) in B '/\' C by PARTIT1:def_4; h . D in D by A16, A33; then ((h . B) /\ (h . C)) /\ (h . D) in INTERSECTION ((B '/\' C),D) by A54, SETFAM_1:def_5; then ((h . B) /\ (h . C)) /\ (h . D) in (INTERSECTION ((B '/\' C),D)) \ {{}} by A41, XBOOLE_0:def_5; then A55: ((h . B) /\ (h . C)) /\ (h . D) in (B '/\' C) '/\' D by PARTIT1:def_4; (((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E) <> {} by A18, A35; then A56: not (((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E) in {{}} by TARSKI:def_1; h . E in E by A16, A21; then (((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E) in INTERSECTION (((B '/\' C) '/\' D),E) by A55, SETFAM_1:def_5; then (((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E) in (INTERSECTION (((B '/\' C) '/\' D),E)) \ {{}} by A56, XBOOLE_0:def_5; then A57: (((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E) in ((B '/\' C) '/\' D) '/\' E by PARTIT1:def_4; ((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F) <> {} by A18, A35; then A58: not ((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F) in {{}} by TARSKI:def_1; h . F in F by A16, A28; then ((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F) in INTERSECTION ((((B '/\' C) '/\' D) '/\' E),F) by A57, SETFAM_1:def_5; then ((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F) in (INTERSECTION ((((B '/\' C) '/\' D) '/\' E),F)) \ {{}} by A58, XBOOLE_0:def_5; then A59: ((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F) in (((B '/\' C) '/\' D) '/\' E) '/\' F by PARTIT1:def_4; h . J in J by A16, A31; then x in INTERSECTION (((((B '/\' C) '/\' D) '/\' E) '/\' F),J) by A53, A59, SETFAM_1:def_5; then x in (INTERSECTION (((((B '/\' C) '/\' D) '/\' E) '/\' F),J)) \ {{}} by A30, XBOOLE_0:def_5; hence x in ((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J by PARTIT1:def_4; ::_thesis: verum end; A60: ((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J c= '/\' (G \ {A}) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in ((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J or x in '/\' (G \ {A}) ) assume A61: x in ((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J ; ::_thesis: x in '/\' (G \ {A}) then A62: x <> {} by EQREL_1:def_4; x in (INTERSECTION (((((B '/\' C) '/\' D) '/\' E) '/\' F),J)) \ {{}} by A61, PARTIT1:def_4; then consider bcdef, j being set such that A63: bcdef in (((B '/\' C) '/\' D) '/\' E) '/\' F and A64: j in J and A65: x = bcdef /\ j by SETFAM_1:def_5; bcdef in (INTERSECTION ((((B '/\' C) '/\' D) '/\' E),F)) \ {{}} by A63, PARTIT1:def_4; then consider bcde, f being set such that A66: bcde in ((B '/\' C) '/\' D) '/\' E and A67: f in F and A68: bcdef = bcde /\ f by SETFAM_1:def_5; bcde in (INTERSECTION (((B '/\' C) '/\' D),E)) \ {{}} by A66, PARTIT1:def_4; then consider bcd, e being set such that A69: bcd in (B '/\' C) '/\' D and A70: e in E and A71: bcde = bcd /\ e by SETFAM_1:def_5; bcd in (INTERSECTION ((B '/\' C),D)) \ {{}} by A69, PARTIT1:def_4; then consider bc, d being set such that A72: bc in B '/\' C and A73: d in D and A74: bcd = bc /\ d by SETFAM_1:def_5; bc in (INTERSECTION (B,C)) \ {{}} by A72, PARTIT1:def_4; then consider b, c being set such that A75: ( b in B & c in C ) and A76: bc = b /\ c by SETFAM_1:def_5; set h = (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j); A77: ((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . B = b by A6, Th37; A78: dom ((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) = {J,B,C,D,E,F} by Th38 .= {J} \/ {B,C,D,E,F} by ENUMSET1:11 .= {B,C,D,E,F,J} by ENUMSET1:15 ; then D in dom ((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) by ENUMSET1:def_4; then A79: ((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . D in rng ((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) by FUNCT_1:def_3; A80: for p being set st p in G \ {A} holds ((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . p in p proof let p be set ; ::_thesis: ( p in G \ {A} implies ((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . p in p ) assume p in G \ {A} ; ::_thesis: ((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . p in p then ( p = B or p = C or p = D or p = E or p = F or p = J ) by A12, ENUMSET1:def_4; hence ((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . p in p by A6, A64, A67, A70, A73, A75, Th37; ::_thesis: verum end; E in dom ((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) by A78, ENUMSET1:def_4; then A81: ((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . E in rng ((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) by FUNCT_1:def_3; C in dom ((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) by A78, ENUMSET1:def_4; then A82: ((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . C in rng ((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) by FUNCT_1:def_3; A83: ((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . C = c by A6, Th37; A84: rng ((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) c= {(((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . B),(((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . C),(((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . D),(((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . E),(((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . F),(((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . J)} proof let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in rng ((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) or t in {(((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . B),(((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . C),(((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . D),(((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . E),(((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . F),(((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . J)} ) assume t in rng ((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) ; ::_thesis: t in {(((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . B),(((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . C),(((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . D),(((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . E),(((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . F),(((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . J)} then consider x1 being set such that A85: x1 in dom ((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) and A86: t = ((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . x1 by FUNCT_1:def_3; ( x1 = B or x1 = C or x1 = D or x1 = E or x1 = F or x1 = J ) by A78, A85, ENUMSET1:def_4; hence t in {(((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . B),(((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . C),(((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . D),(((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . E),(((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . F),(((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . J)} by A86, ENUMSET1:def_4; ::_thesis: verum end; J in dom ((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) by A78, ENUMSET1:def_4; then A87: ((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . J in rng ((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) by FUNCT_1:def_3; F in dom ((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) by A78, ENUMSET1:def_4; then A88: ((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . F in rng ((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) by FUNCT_1:def_3; B in dom ((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) by A78, ENUMSET1:def_4; then A89: ((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . B in rng ((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) by FUNCT_1:def_3; {(((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . B),(((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . C),(((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . D),(((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . E),(((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . F),(((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . J)} c= rng ((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) proof let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in {(((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . B),(((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . C),(((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . D),(((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . E),(((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . F),(((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . J)} or t in rng ((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) ) assume t in {(((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . B),(((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . C),(((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . D),(((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . E),(((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . F),(((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . J)} ; ::_thesis: t in rng ((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) hence t in rng ((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) by A79, A89, A82, A81, A88, A87, ENUMSET1:def_4; ::_thesis: verum end; then A90: rng ((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) = {(((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . B),(((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . C),(((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . D),(((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . E),(((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . F),(((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . J)} by A84, XBOOLE_0:def_10; A91: ((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . J = j by A6, Th37; A92: ((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . F = f by A6, Th37; A93: ((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . E = e by A6, Th37; A94: ((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . D = d by A6, Th37; rng ((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) c= bool Y proof let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in rng ((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) or t in bool Y ) assume t in rng ((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) ; ::_thesis: t in bool Y then ( t = ((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . D or t = ((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . B or t = ((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . C or t = ((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . E or t = ((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . F or t = ((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) . J ) by A84, ENUMSET1:def_4; hence t in bool Y by A64, A67, A70, A73, A75, A94, A77, A83, A93, A92, A91; ::_thesis: verum end; then reconsider FF = rng ((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) as Subset-Family of Y ; A95: dom ((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) = {J,B,C,D,E,F} by Th38 .= {J} \/ {B,C,D,E,F} by ENUMSET1:11 .= {B,C,D,E,F,J} by ENUMSET1:15 ; reconsider h = (((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j) as Function ; A96: x c= Intersect FF proof let u be set ; :: according to TARSKI:def_3 ::_thesis: ( not u in x or u in Intersect FF ) assume A97: u in x ; ::_thesis: u in Intersect FF for y being set st y in FF holds u in y proof let y be set ; ::_thesis: ( y in FF implies u in y ) assume A98: y in FF ; ::_thesis: u in y now__::_thesis:_(_(_y_=_h_._D_&_u_in_y_)_or_(_y_=_h_._B_&_u_in_y_)_or_(_y_=_h_._C_&_u_in_y_)_or_(_y_=_h_._E_&_u_in_y_)_or_(_y_=_h_._F_&_u_in_y_)_or_(_y_=_h_._J_&_u_in_y_)_) percases ( y = h . D or y = h . B or y = h . C or y = h . E or y = h . F or y = h . J ) by A84, A98, ENUMSET1:def_4; caseA99: y = h . D ; ::_thesis: u in y u in ((d /\ ((b /\ c) /\ e)) /\ f) /\ j by A65, A68, A71, A74, A76, A97, XBOOLE_1:16; then u in (d /\ (((b /\ c) /\ e) /\ f)) /\ j by XBOOLE_1:16; then u in d /\ ((((b /\ c) /\ e) /\ f) /\ j) by XBOOLE_1:16; hence u in y by A94, A99, XBOOLE_0:def_4; ::_thesis: verum end; caseA100: y = h . B ; ::_thesis: u in y u in (((c /\ (d /\ b)) /\ e) /\ f) /\ j by A65, A68, A71, A74, A76, A97, XBOOLE_1:16; then u in ((c /\ ((d /\ b) /\ e)) /\ f) /\ j by XBOOLE_1:16; then u in ((c /\ ((d /\ e) /\ b)) /\ f) /\ j by XBOOLE_1:16; then u in (c /\ (((d /\ e) /\ b) /\ f)) /\ j by XBOOLE_1:16; then u in c /\ ((((d /\ e) /\ b) /\ f) /\ j) by XBOOLE_1:16; then u in c /\ (((d /\ e) /\ (f /\ b)) /\ j) by XBOOLE_1:16; then u in c /\ ((d /\ e) /\ ((f /\ b) /\ j)) by XBOOLE_1:16; then u in c /\ ((d /\ e) /\ (f /\ (j /\ b))) by XBOOLE_1:16; then u in (c /\ (d /\ e)) /\ (f /\ (j /\ b)) by XBOOLE_1:16; then u in ((c /\ (d /\ e)) /\ f) /\ (j /\ b) by XBOOLE_1:16; then u in (((c /\ (d /\ e)) /\ f) /\ j) /\ b by XBOOLE_1:16; hence u in y by A77, A100, XBOOLE_0:def_4; ::_thesis: verum end; caseA101: y = h . C ; ::_thesis: u in y u in (((c /\ (d /\ b)) /\ e) /\ f) /\ j by A65, A68, A71, A74, A76, A97, XBOOLE_1:16; then u in ((c /\ ((d /\ b) /\ e)) /\ f) /\ j by XBOOLE_1:16; then u in ((c /\ ((d /\ e) /\ b)) /\ f) /\ j by XBOOLE_1:16; then u in (c /\ (((d /\ e) /\ b) /\ f)) /\ j by XBOOLE_1:16; then u in c /\ ((((d /\ e) /\ b) /\ f) /\ j) by XBOOLE_1:16; hence u in y by A83, A101, XBOOLE_0:def_4; ::_thesis: verum end; caseA102: y = h . E ; ::_thesis: u in y u in (((b /\ c) /\ d) /\ (f /\ e)) /\ j by A65, A68, A71, A74, A76, A97, XBOOLE_1:16; then u in ((b /\ c) /\ d) /\ ((f /\ e) /\ j) by XBOOLE_1:16; then u in ((b /\ c) /\ d) /\ ((f /\ j) /\ e) by XBOOLE_1:16; then u in (((b /\ c) /\ d) /\ (f /\ j)) /\ e by XBOOLE_1:16; hence u in y by A93, A102, XBOOLE_0:def_4; ::_thesis: verum end; caseA103: y = h . F ; ::_thesis: u in y u in ((((b /\ c) /\ d) /\ e) /\ j) /\ f by A65, A68, A71, A74, A76, A97, XBOOLE_1:16; hence u in y by A92, A103, XBOOLE_0:def_4; ::_thesis: verum end; case y = h . J ; ::_thesis: u in y hence u in y by A65, A91, A97, XBOOLE_0:def_4; ::_thesis: verum end; end; end; hence u in y ; ::_thesis: verum end; then u in meet FF by A90, SETFAM_1:def_1; hence u in Intersect FF by A90, SETFAM_1:def_9; ::_thesis: verum end; A104: Intersect FF = meet (rng h) by A79, SETFAM_1:def_9; Intersect FF c= x proof let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in Intersect FF or t in x ) assume A105: t in Intersect FF ; ::_thesis: t in x h . C in rng h by A90, ENUMSET1:def_4; then A106: t in c by A83, A104, A105, SETFAM_1:def_1; h . B in rng h by A90, ENUMSET1:def_4; then t in b by A77, A104, A105, SETFAM_1:def_1; then A107: t in b /\ c by A106, XBOOLE_0:def_4; h . D in rng h by A90, ENUMSET1:def_4; then t in d by A94, A104, A105, SETFAM_1:def_1; then A108: t in (b /\ c) /\ d by A107, XBOOLE_0:def_4; h . E in rng h by A90, ENUMSET1:def_4; then t in e by A93, A104, A105, SETFAM_1:def_1; then A109: t in ((b /\ c) /\ d) /\ e by A108, XBOOLE_0:def_4; h . F in rng h by A90, ENUMSET1:def_4; then t in f by A92, A104, A105, SETFAM_1:def_1; then A110: t in (((b /\ c) /\ d) /\ e) /\ f by A109, XBOOLE_0:def_4; h . J in rng h by A90, ENUMSET1:def_4; then t in j by A91, A104, A105, SETFAM_1:def_1; hence t in x by A65, A68, A71, A74, A76, A110, XBOOLE_0:def_4; ::_thesis: verum end; then x = Intersect FF by A96, XBOOLE_0:def_10; hence x in '/\' (G \ {A}) by A12, A95, A80, A62, BVFUNC_2:def_1; ::_thesis: verum end; CompF (A,G) = '/\' (G \ {A}) by BVFUNC_2:def_7; hence CompF (A,G) = ((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J by A60, A13, XBOOLE_0:def_10; ::_thesis: verum end; theorem Th43: :: BVFUNC14:43 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J being a_partition of Y st G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J holds CompF (B,G) = ((((A '/\' C) '/\' D) '/\' E) '/\' F) '/\' J proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J being a_partition of Y st G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J holds CompF (B,G) = ((((A '/\' C) '/\' D) '/\' E) '/\' F) '/\' J let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C, D, E, F, J being a_partition of Y st G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J holds CompF (B,G) = ((((A '/\' C) '/\' D) '/\' E) '/\' F) '/\' J let A, B, C, D, E, F, J be a_partition of Y; ::_thesis: ( G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J implies CompF (B,G) = ((((A '/\' C) '/\' D) '/\' E) '/\' F) '/\' J ) {A,B,C,D,E,F,J} = {A,B} \/ {C,D,E,F,J} by ENUMSET1:17 .= {B,A,C,D,E,F,J} by ENUMSET1:17 ; hence ( G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J implies CompF (B,G) = ((((A '/\' C) '/\' D) '/\' E) '/\' F) '/\' J ) by Th42; ::_thesis: verum end; theorem Th44: :: BVFUNC14:44 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J being a_partition of Y st G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J holds CompF (C,G) = ((((A '/\' B) '/\' D) '/\' E) '/\' F) '/\' J proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J being a_partition of Y st G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J holds CompF (C,G) = ((((A '/\' B) '/\' D) '/\' E) '/\' F) '/\' J let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C, D, E, F, J being a_partition of Y st G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J holds CompF (C,G) = ((((A '/\' B) '/\' D) '/\' E) '/\' F) '/\' J let A, B, C, D, E, F, J be a_partition of Y; ::_thesis: ( G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J implies CompF (C,G) = ((((A '/\' B) '/\' D) '/\' E) '/\' F) '/\' J ) {A,B,C,D,E,F,J} = {A,B,C} \/ {D,E,F,J} by ENUMSET1:18 .= ({A} \/ {B,C}) \/ {D,E,F,J} by ENUMSET1:2 .= {A,C,B} \/ {D,E,F,J} by ENUMSET1:2 .= ({A,C} \/ {B}) \/ {D,E,F,J} by ENUMSET1:3 .= {C,A,B} \/ {D,E,F,J} by ENUMSET1:3 .= {C,A,B,D,E,F,J} by ENUMSET1:18 ; hence ( G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J implies CompF (C,G) = ((((A '/\' B) '/\' D) '/\' E) '/\' F) '/\' J ) by Th42; ::_thesis: verum end; theorem Th45: :: BVFUNC14:45 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J being a_partition of Y st G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J holds CompF (D,G) = ((((A '/\' B) '/\' C) '/\' E) '/\' F) '/\' J proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J being a_partition of Y st G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J holds CompF (D,G) = ((((A '/\' B) '/\' C) '/\' E) '/\' F) '/\' J let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C, D, E, F, J being a_partition of Y st G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J holds CompF (D,G) = ((((A '/\' B) '/\' C) '/\' E) '/\' F) '/\' J let A, B, C, D, E, F, J be a_partition of Y; ::_thesis: ( G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J implies CompF (D,G) = ((((A '/\' B) '/\' C) '/\' E) '/\' F) '/\' J ) {A,B,C,D,E,F,J} = {A,B} \/ {C,D,E,F,J} by ENUMSET1:17 .= {A,B} \/ ({C,D} \/ {E,F,J}) by ENUMSET1:8 .= {A,B} \/ {D,C,E,F,J} by ENUMSET1:8 .= {A,B,D,C,E,F,J} by ENUMSET1:17 ; hence ( G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J implies CompF (D,G) = ((((A '/\' B) '/\' C) '/\' E) '/\' F) '/\' J ) by Th44; ::_thesis: verum end; theorem Th46: :: BVFUNC14:46 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J being a_partition of Y st G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J holds CompF (E,G) = ((((A '/\' B) '/\' C) '/\' D) '/\' F) '/\' J proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J being a_partition of Y st G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J holds CompF (E,G) = ((((A '/\' B) '/\' C) '/\' D) '/\' F) '/\' J let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C, D, E, F, J being a_partition of Y st G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J holds CompF (E,G) = ((((A '/\' B) '/\' C) '/\' D) '/\' F) '/\' J let A, B, C, D, E, F, J be a_partition of Y; ::_thesis: ( G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J implies CompF (E,G) = ((((A '/\' B) '/\' C) '/\' D) '/\' F) '/\' J ) {A,B,C,D,E,F,J} = {A,B,C} \/ {D,E,F,J} by ENUMSET1:18 .= {A,B,C} \/ ({D,E} \/ {F,J}) by ENUMSET1:5 .= {A,B,C} \/ {E,D,F,J} by ENUMSET1:5 .= {A,B,C,E,D,F,J} by ENUMSET1:18 ; hence ( G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J implies CompF (E,G) = ((((A '/\' B) '/\' C) '/\' D) '/\' F) '/\' J ) by Th45; ::_thesis: verum end; theorem Th47: :: BVFUNC14:47 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J being a_partition of Y st G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J holds CompF (F,G) = ((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' J proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J being a_partition of Y st G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J holds CompF (F,G) = ((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' J let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C, D, E, F, J being a_partition of Y st G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J holds CompF (F,G) = ((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' J let A, B, C, D, E, F, J be a_partition of Y; ::_thesis: ( G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J implies CompF (F,G) = ((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' J ) {A,B,C,D,E,F,J} = {A,B,C,D} \/ {E,F,J} by ENUMSET1:19 .= {A,B,C,D} \/ {F,E,J} by ENUMSET1:58 .= {A,B,C,D,F,E,J} by ENUMSET1:19 ; hence ( G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J implies CompF (F,G) = ((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' J ) by Th46; ::_thesis: verum end; theorem :: BVFUNC14:48 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J being a_partition of Y st G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J holds CompF (J,G) = ((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' F proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J being a_partition of Y st G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J holds CompF (J,G) = ((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' F let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C, D, E, F, J being a_partition of Y st G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J holds CompF (J,G) = ((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' F let A, B, C, D, E, F, J be a_partition of Y; ::_thesis: ( G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J implies CompF (J,G) = ((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' F ) {A,B,C,D,E,F,J} = {A,B,C,D,E} \/ {F,J} by ENUMSET1:20 .= {A,B,C,D,E,J,F} by ENUMSET1:20 ; hence ( G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J implies CompF (J,G) = ((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' F ) by Th47; ::_thesis: verum end; theorem Th49: :: BVFUNC14:49 for A, B, C, D, E, F, J being set for h being Function for A9, B9, C9, D9, E9, F9, J9 being set st A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J & h = ((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (A .--> A9) holds ( h . A = A9 & h . B = B9 & h . C = C9 & h . D = D9 & h . E = E9 & h . F = F9 & h . J = J9 ) proof let A, B, C, D, E, F, J be set ; ::_thesis: for h being Function for A9, B9, C9, D9, E9, F9, J9 being set st A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J & h = ((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (A .--> A9) holds ( h . A = A9 & h . B = B9 & h . C = C9 & h . D = D9 & h . E = E9 & h . F = F9 & h . J = J9 ) let h be Function; ::_thesis: for A9, B9, C9, D9, E9, F9, J9 being set st A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J & h = ((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (A .--> A9) holds ( h . A = A9 & h . B = B9 & h . C = C9 & h . D = D9 & h . E = E9 & h . F = F9 & h . J = J9 ) let A9, B9, C9, D9, E9, F9, J9 be set ; ::_thesis: ( A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J & h = ((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (A .--> A9) implies ( h . A = A9 & h . B = B9 & h . C = C9 & h . D = D9 & h . E = E9 & h . F = F9 & h . J = J9 ) ) assume that A1: A <> B and A2: A <> C and A3: A <> D and A4: A <> E and A5: A <> F and A6: A <> J and A7: ( B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J ) and A8: h = ((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (A .--> A9) ; ::_thesis: ( h . A = A9 & h . B = B9 & h . C = C9 & h . D = D9 & h . E = E9 & h . F = F9 & h . J = J9 ) A9: dom (A .--> A9) = {A} by FUNCOP_1:13; then A in dom (A .--> A9) by TARSKI:def_1; then A10: h . A = (A .--> A9) . A by A8, FUNCT_4:13; not J in dom (A .--> A9) by A6, A9, TARSKI:def_1; then A11: h . J = ((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) . J by A8, FUNCT_4:11 .= J9 by A7, Th37 ; not F in dom (A .--> A9) by A5, A9, TARSKI:def_1; then A12: h . F = ((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) . F by A8, FUNCT_4:11 .= F9 by A7, Th37 ; not E in dom (A .--> A9) by A4, A9, TARSKI:def_1; then A13: h . E = ((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) . E by A8, FUNCT_4:11 .= E9 by A7, Th37 ; not D in dom (A .--> A9) by A3, A9, TARSKI:def_1; then A14: h . D = ((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) . D by A8, FUNCT_4:11 .= D9 by A7, Th37 ; not C in dom (A .--> A9) by A2, A9, TARSKI:def_1; then A15: h . C = ((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) . C by A8, FUNCT_4:11 .= C9 by A7, Th37 ; not B in dom (A .--> A9) by A1, A9, TARSKI:def_1; then h . B = ((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) . B by A8, FUNCT_4:11 .= B9 by A7, Th37 ; hence ( h . A = A9 & h . B = B9 & h . C = C9 & h . D = D9 & h . E = E9 & h . F = F9 & h . J = J9 ) by A10, A15, A14, A13, A12, A11, FUNCOP_1:72; ::_thesis: verum end; theorem Th50: :: BVFUNC14:50 for A, B, C, D, E, F, J being set for h being Function for A9, B9, C9, D9, E9, F9, J9 being set st h = ((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (A .--> A9) holds dom h = {A,B,C,D,E,F,J} proof let A, B, C, D, E, F, J be set ; ::_thesis: for h being Function for A9, B9, C9, D9, E9, F9, J9 being set st h = ((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (A .--> A9) holds dom h = {A,B,C,D,E,F,J} let h be Function; ::_thesis: for A9, B9, C9, D9, E9, F9, J9 being set st h = ((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (A .--> A9) holds dom h = {A,B,C,D,E,F,J} let A9, B9, C9, D9, E9, F9, J9 be set ; ::_thesis: ( h = ((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (A .--> A9) implies dom h = {A,B,C,D,E,F,J} ) A1: dom (A .--> A9) = {A} by FUNCOP_1:13; assume h = ((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (A .--> A9) ; ::_thesis: dom h = {A,B,C,D,E,F,J} then dom h = (dom ((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9))) \/ (dom (A .--> A9)) by FUNCT_4:def_1 .= {J,B,C,D,E,F} \/ (dom (A .--> A9)) by Th38 .= ({B,C,D,E,F} \/ {J}) \/ {A} by A1, ENUMSET1:11 .= {B,C,D,E,F,J} \/ {A} by ENUMSET1:15 .= {A,B,C,D,E,F,J} by ENUMSET1:16 ; hence dom h = {A,B,C,D,E,F,J} ; ::_thesis: verum end; theorem Th51: :: BVFUNC14:51 for A, B, C, D, E, F, J being set for h being Function for A9, B9, C9, D9, E9, F9, J9 being set st h = ((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (A .--> A9) holds rng h = {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J)} proof let A, B, C, D, E, F, J be set ; ::_thesis: for h being Function for A9, B9, C9, D9, E9, F9, J9 being set st h = ((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (A .--> A9) holds rng h = {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J)} let h be Function; ::_thesis: for A9, B9, C9, D9, E9, F9, J9 being set st h = ((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (A .--> A9) holds rng h = {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J)} let A9, B9, C9, D9, E9, F9, J9 be set ; ::_thesis: ( h = ((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (A .--> A9) implies rng h = {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J)} ) assume h = ((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (A .--> A9) ; ::_thesis: rng h = {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J)} then A1: dom h = {A,B,C,D,E,F,J} by Th50; then B in dom h by ENUMSET1:def_5; then A2: h . B in rng h by FUNCT_1:def_3; F in dom h by A1, ENUMSET1:def_5; then A3: h . F in rng h by FUNCT_1:def_3; E in dom h by A1, ENUMSET1:def_5; then A4: h . E in rng h by FUNCT_1:def_3; D in dom h by A1, ENUMSET1:def_5; then A5: h . D in rng h by FUNCT_1:def_3; C in dom h by A1, ENUMSET1:def_5; then A6: h . C in rng h by FUNCT_1:def_3; J in dom h by A1, ENUMSET1:def_5; then A7: h . J in rng h by FUNCT_1:def_3; A8: rng h c= {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J)} proof let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in rng h or t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J)} ) assume t in rng h ; ::_thesis: t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J)} then consider x1 being set such that A9: x1 in dom h and A10: t = h . x1 by FUNCT_1:def_3; ( x1 = A or x1 = B or x1 = C or x1 = D or x1 = E or x1 = F or x1 = J ) by A1, A9, ENUMSET1:def_5; hence t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J)} by A10, ENUMSET1:def_5; ::_thesis: verum end; A in dom h by A1, ENUMSET1:def_5; then A11: h . A in rng h by FUNCT_1:def_3; {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J)} c= rng h proof let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J)} or t in rng h ) assume t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J)} ; ::_thesis: t in rng h hence t in rng h by A11, A2, A6, A5, A4, A3, A7, ENUMSET1:def_5; ::_thesis: verum end; hence rng h = {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J)} by A8, XBOOLE_0:def_10; ::_thesis: verum end; theorem :: BVFUNC14:52 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J being a_partition of Y for z, u being Element of Y st G is independent & G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J holds EqClass (u,(((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J)) meets EqClass (z,A) proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J being a_partition of Y for z, u being Element of Y st G is independent & G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J holds EqClass (u,(((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J)) meets EqClass (z,A) let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C, D, E, F, J being a_partition of Y for z, u being Element of Y st G is independent & G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J holds EqClass (u,(((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J)) meets EqClass (z,A) let A, B, C, D, E, F, J be a_partition of Y; ::_thesis: for z, u being Element of Y st G is independent & G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J holds EqClass (u,(((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J)) meets EqClass (z,A) let z, u be Element of Y; ::_thesis: ( G is independent & G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J implies EqClass (u,(((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J)) meets EqClass (z,A) ) assume that A1: G is independent and A2: G = {A,B,C,D,E,F,J} and A3: ( A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J ) ; ::_thesis: EqClass (u,(((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J)) meets EqClass (z,A) set h = ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A))); A4: (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . A = EqClass (z,A) by A3, Th49; reconsider GG = EqClass (u,(((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J)) as set ; reconsider I = EqClass (z,A) as set ; GG = (EqClass (u,((((B '/\' C) '/\' D) '/\' E) '/\' F))) /\ (EqClass (u,J)) by Th1; then GG = ((EqClass (u,(((B '/\' C) '/\' D) '/\' E))) /\ (EqClass (u,F))) /\ (EqClass (u,J)) by Th1; then GG = (((EqClass (u,((B '/\' C) '/\' D))) /\ (EqClass (u,E))) /\ (EqClass (u,F))) /\ (EqClass (u,J)) by Th1; then GG = ((((EqClass (u,(B '/\' C))) /\ (EqClass (u,D))) /\ (EqClass (u,E))) /\ (EqClass (u,F))) /\ (EqClass (u,J)) by Th1; then A5: GG /\ I = ((((((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E))) /\ (EqClass (u,F))) /\ (EqClass (u,J))) /\ (EqClass (z,A)) by Th1; A6: (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . B = EqClass (u,B) by A3, Th49; A7: (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . F = EqClass (u,F) by A3, Th49; A8: (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . E = EqClass (u,E) by A3, Th49; A9: (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . J = EqClass (u,J) by A3, Th49; A10: (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . D = EqClass (u,D) by A3, Th49; A11: (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . C = EqClass (u,C) by A3, Th49; A12: rng (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) = {((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . A),((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . B),((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . C),((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . D),((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . E),((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . F),((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . J)} by Th51; rng (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) c= bool Y proof let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in rng (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) or t in bool Y ) assume t in rng (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) ; ::_thesis: t in bool Y then ( t = (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . A or t = (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . B or t = (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . C or t = (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . D or t = (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . E or t = (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . F or t = (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . J ) by A12, ENUMSET1:def_5; hence t in bool Y by A4, A6, A11, A10, A8, A7, A9; ::_thesis: verum end; then reconsider FF = rng (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) as Subset-Family of Y ; A13: dom (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) = G by A2, Th50; then A in dom (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) by A2, ENUMSET1:def_5; then A14: (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . A in rng (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then A15: Intersect FF = meet (rng (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A))))) by SETFAM_1:def_9; for d being set st d in G holds (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . d in d proof let d be set ; ::_thesis: ( d in G implies (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . d in d ) assume d in G ; ::_thesis: (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . d in d then ( d = A or d = B or d = C or d = D or d = E or d = F or d = J ) by A2, ENUMSET1:def_5; hence (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . d in d by A4, A6, A11, A10, A8, A7, A9; ::_thesis: verum end; then Intersect FF <> {} by A1, A13, BVFUNC_2:def_5; then consider m being set such that A16: m in Intersect FF by XBOOLE_0:def_1; C in dom (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) by A2, A13, ENUMSET1:def_5; then (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . C in rng (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then A17: m in EqClass (u,C) by A11, A15, A16, SETFAM_1:def_1; B in dom (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) by A2, A13, ENUMSET1:def_5; then (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . B in rng (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then m in EqClass (u,B) by A6, A15, A16, SETFAM_1:def_1; then A18: m in (EqClass (u,B)) /\ (EqClass (u,C)) by A17, XBOOLE_0:def_4; D in dom (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) by A2, A13, ENUMSET1:def_5; then (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . D in rng (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then m in EqClass (u,D) by A10, A15, A16, SETFAM_1:def_1; then A19: m in ((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D)) by A18, XBOOLE_0:def_4; E in dom (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) by A2, A13, ENUMSET1:def_5; then (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . E in rng (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then m in EqClass (u,E) by A8, A15, A16, SETFAM_1:def_1; then A20: m in (((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E)) by A19, XBOOLE_0:def_4; F in dom (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) by A2, A13, ENUMSET1:def_5; then (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . F in rng (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then m in EqClass (u,F) by A7, A15, A16, SETFAM_1:def_1; then A21: m in ((((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E))) /\ (EqClass (u,F)) by A20, XBOOLE_0:def_4; J in dom (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) by A2, A13, ENUMSET1:def_5; then (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . J in rng (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then m in EqClass (u,J) by A9, A15, A16, SETFAM_1:def_1; then A22: m in (((((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E))) /\ (EqClass (u,F))) /\ (EqClass (u,J)) by A21, XBOOLE_0:def_4; m in EqClass (z,A) by A4, A14, A15, A16, SETFAM_1:def_1; then m in (EqClass (u,(((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J))) /\ (EqClass (z,A)) by A5, A22, XBOOLE_0:def_4; hence EqClass (u,(((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J)) meets EqClass (z,A) by XBOOLE_0:def_7; ::_thesis: verum end; theorem :: BVFUNC14:53 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J being a_partition of Y for z, u being Element of Y st G is independent & G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J & EqClass (z,((((C '/\' D) '/\' E) '/\' F) '/\' J)) = EqClass (u,((((C '/\' D) '/\' E) '/\' F) '/\' J)) holds EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G))) proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J being a_partition of Y for z, u being Element of Y st G is independent & G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J & EqClass (z,((((C '/\' D) '/\' E) '/\' F) '/\' J)) = EqClass (u,((((C '/\' D) '/\' E) '/\' F) '/\' J)) holds EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G))) let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C, D, E, F, J being a_partition of Y for z, u being Element of Y st G is independent & G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J & EqClass (z,((((C '/\' D) '/\' E) '/\' F) '/\' J)) = EqClass (u,((((C '/\' D) '/\' E) '/\' F) '/\' J)) holds EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G))) let A, B, C, D, E, F, J be a_partition of Y; ::_thesis: for z, u being Element of Y st G is independent & G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J & EqClass (z,((((C '/\' D) '/\' E) '/\' F) '/\' J)) = EqClass (u,((((C '/\' D) '/\' E) '/\' F) '/\' J)) holds EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G))) let z, u be Element of Y; ::_thesis: ( G is independent & G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J & EqClass (z,((((C '/\' D) '/\' E) '/\' F) '/\' J)) = EqClass (u,((((C '/\' D) '/\' E) '/\' F) '/\' J)) implies EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G))) ) assume that A1: G is independent and A2: G = {A,B,C,D,E,F,J} and A3: ( A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J ) and A4: EqClass (z,((((C '/\' D) '/\' E) '/\' F) '/\' J)) = EqClass (u,((((C '/\' D) '/\' E) '/\' F) '/\' J)) ; ::_thesis: EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G))) set h = ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A))); A5: (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . A = EqClass (z,A) by A3, Th49; reconsider L = EqClass (z,((((C '/\' D) '/\' E) '/\' F) '/\' J)) as set ; reconsider GG = EqClass (u,(((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J)) as set ; reconsider I = EqClass (z,A) as set ; GG = (EqClass (u,((((B '/\' C) '/\' D) '/\' E) '/\' F))) /\ (EqClass (u,J)) by Th1; then GG = ((EqClass (u,(((B '/\' C) '/\' D) '/\' E))) /\ (EqClass (u,F))) /\ (EqClass (u,J)) by Th1; then GG = (((EqClass (u,((B '/\' C) '/\' D))) /\ (EqClass (u,E))) /\ (EqClass (u,F))) /\ (EqClass (u,J)) by Th1; then GG = ((((EqClass (u,(B '/\' C))) /\ (EqClass (u,D))) /\ (EqClass (u,E))) /\ (EqClass (u,F))) /\ (EqClass (u,J)) by Th1; then A6: GG /\ I = ((((((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E))) /\ (EqClass (u,F))) /\ (EqClass (u,J))) /\ (EqClass (z,A)) by Th1; A7: CompF (A,G) = ((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J by A2, A3, Th42; reconsider HH = EqClass (z,(CompF (B,G))) as set ; A8: z in HH by EQREL_1:def_6; A9: A '/\' ((((C '/\' D) '/\' E) '/\' F) '/\' J) = (A '/\' (((C '/\' D) '/\' E) '/\' F)) '/\' J by PARTIT1:14 .= ((A '/\' ((C '/\' D) '/\' E)) '/\' F) '/\' J by PARTIT1:14 .= (((A '/\' (C '/\' D)) '/\' E) '/\' F) '/\' J by PARTIT1:14 .= ((((A '/\' C) '/\' D) '/\' E) '/\' F) '/\' J by PARTIT1:14 ; A10: (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . B = EqClass (u,B) by A3, Th49; A11: (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . F = EqClass (u,F) by A3, Th49; A12: (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . E = EqClass (u,E) by A3, Th49; A13: (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . J = EqClass (u,J) by A3, Th49; A14: (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . D = EqClass (u,D) by A3, Th49; A15: (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . C = EqClass (u,C) by A3, Th49; A16: rng (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) = {((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . A),((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . B),((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . C),((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . D),((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . E),((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . F),((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . J)} by Th51; rng (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) c= bool Y proof let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in rng (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) or t in bool Y ) assume t in rng (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) ; ::_thesis: t in bool Y then ( t = (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . A or t = (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . B or t = (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . C or t = (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . D or t = (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . E or t = (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . F or t = (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . J ) by A16, ENUMSET1:def_5; hence t in bool Y by A5, A10, A15, A14, A12, A11, A13; ::_thesis: verum end; then reconsider FF = rng (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) as Subset-Family of Y ; A17: dom (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) = G by A2, Th50; then A in dom (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) by A2, ENUMSET1:def_5; then A18: (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . A in rng (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then A19: Intersect FF = meet (rng (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A))))) by SETFAM_1:def_9; for d being set st d in G holds (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . d in d proof let d be set ; ::_thesis: ( d in G implies (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . d in d ) assume d in G ; ::_thesis: (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . d in d then ( d = A or d = B or d = C or d = D or d = E or d = F or d = J ) by A2, ENUMSET1:def_5; hence (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . d in d by A5, A10, A15, A14, A12, A11, A13; ::_thesis: verum end; then Intersect FF <> {} by A1, A17, BVFUNC_2:def_5; then consider m being set such that A20: m in Intersect FF by XBOOLE_0:def_1; C in dom (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) by A2, A17, ENUMSET1:def_5; then (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . C in rng (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then A21: m in EqClass (u,C) by A15, A19, A20, SETFAM_1:def_1; B in dom (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) by A2, A17, ENUMSET1:def_5; then (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . B in rng (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then m in EqClass (u,B) by A10, A19, A20, SETFAM_1:def_1; then A22: m in (EqClass (u,B)) /\ (EqClass (u,C)) by A21, XBOOLE_0:def_4; D in dom (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) by A2, A17, ENUMSET1:def_5; then (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . D in rng (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then m in EqClass (u,D) by A14, A19, A20, SETFAM_1:def_1; then A23: m in ((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D)) by A22, XBOOLE_0:def_4; E in dom (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) by A2, A17, ENUMSET1:def_5; then (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . E in rng (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then m in EqClass (u,E) by A12, A19, A20, SETFAM_1:def_1; then A24: m in (((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E)) by A23, XBOOLE_0:def_4; F in dom (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) by A2, A17, ENUMSET1:def_5; then (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . F in rng (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then m in EqClass (u,F) by A11, A19, A20, SETFAM_1:def_1; then A25: m in ((((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E))) /\ (EqClass (u,F)) by A24, XBOOLE_0:def_4; J in dom (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) by A2, A17, ENUMSET1:def_5; then (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) . J in rng (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then m in EqClass (u,J) by A13, A19, A20, SETFAM_1:def_1; then A26: m in (((((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E))) /\ (EqClass (u,F))) /\ (EqClass (u,J)) by A25, XBOOLE_0:def_4; m in EqClass (z,A) by A5, A18, A19, A20, SETFAM_1:def_1; then A27: GG /\ I <> {} by A6, A26, XBOOLE_0:def_4; then consider p being set such that A28: p in GG /\ I by XBOOLE_0:def_1; ( GG /\ I in INTERSECTION (A,(((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J)) & not GG /\ I in {{}} ) by A27, SETFAM_1:def_5, TARSKI:def_1; then GG /\ I in (INTERSECTION (A,(((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J))) \ {{}} by XBOOLE_0:def_5; then GG /\ I in A '/\' (((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) by PARTIT1:def_4; then reconsider p = p as Element of Y by A28; A29: p in GG by A28, XBOOLE_0:def_4; reconsider K = EqClass (p,((((C '/\' D) '/\' E) '/\' F) '/\' J)) as set ; A30: p in EqClass (p,((((C '/\' D) '/\' E) '/\' F) '/\' J)) by EQREL_1:def_6; GG = EqClass (u,((((B '/\' (C '/\' D)) '/\' E) '/\' F) '/\' J)) by PARTIT1:14; then GG = EqClass (u,(((B '/\' ((C '/\' D) '/\' E)) '/\' F) '/\' J)) by PARTIT1:14; then GG = EqClass (u,((B '/\' (((C '/\' D) '/\' E) '/\' F)) '/\' J)) by PARTIT1:14; then GG = EqClass (u,(B '/\' ((((C '/\' D) '/\' E) '/\' F) '/\' J))) by PARTIT1:14; then GG c= L by A4, BVFUNC11:3; then K meets L by A29, A30, XBOOLE_0:3; then K = L by EQREL_1:41; then A31: z in K by EQREL_1:def_6; ( p in K & p in I ) by A28, EQREL_1:def_6, XBOOLE_0:def_4; then A32: p in I /\ K by XBOOLE_0:def_4; then ( I /\ K in INTERSECTION (A,((((C '/\' D) '/\' E) '/\' F) '/\' J)) & not I /\ K in {{}} ) by SETFAM_1:def_5, TARSKI:def_1; then I /\ K in (INTERSECTION (A,((((C '/\' D) '/\' E) '/\' F) '/\' J))) \ {{}} by XBOOLE_0:def_5; then A33: I /\ K in A '/\' ((((C '/\' D) '/\' E) '/\' F) '/\' J) by PARTIT1:def_4; z in I by EQREL_1:def_6; then z in I /\ K by A31, XBOOLE_0:def_4; then A34: I /\ K meets HH by A8, XBOOLE_0:3; CompF (B,G) = ((((A '/\' C) '/\' D) '/\' E) '/\' F) '/\' J by A2, A3, Th43; then p in HH by A32, A33, A34, A9, EQREL_1:def_4; hence EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G))) by A7, A29, XBOOLE_0:3; ::_thesis: verum end; theorem Th54: :: BVFUNC14:54 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M being a_partition of Y st G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M holds CompF (A,G) = (((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M being a_partition of Y st G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M holds CompF (A,G) = (((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C, D, E, F, J, M being a_partition of Y st G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M holds CompF (A,G) = (((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M let A, B, C, D, E, F, J, M be a_partition of Y; ::_thesis: ( G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M implies CompF (A,G) = (((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M ) assume that A1: G = {A,B,C,D,E,F,J,M} and A2: A <> B and A3: A <> C and A4: ( A <> D & A <> E ) and A5: ( A <> F & A <> J ) and A6: A <> M and A7: ( B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M ) ; ::_thesis: CompF (A,G) = (((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M A8: not B in {A} by A2, TARSKI:def_1; G \ {A} = ({A} \/ {B,C,D,E,F,J,M}) \ {A} by A1, ENUMSET1:22; then A9: G \ {A} = ({A} \ {A}) \/ ({B,C,D,E,F,J,M} \ {A}) by XBOOLE_1:42; A10: ( not D in {A} & not E in {A} ) by A4, TARSKI:def_1; A11: not C in {A} by A3, TARSKI:def_1; A12: not M in {A} by A6, TARSKI:def_1; A13: ( not F in {A} & not J in {A} ) by A5, TARSKI:def_1; {B,C,D,E,F,J,M} \ {A} = ({B} \/ {C,D,E,F,J,M}) \ {A} by ENUMSET1:16 .= ({B} \ {A}) \/ ({C,D,E,F,J,M} \ {A}) by XBOOLE_1:42 .= {B} \/ ({C,D,E,F,J,M} \ {A}) by A8, ZFMISC_1:59 .= {B} \/ (({C} \/ {D,E,F,J,M}) \ {A}) by ENUMSET1:11 .= {B} \/ (({C} \ {A}) \/ ({D,E,F,J,M} \ {A})) by XBOOLE_1:42 .= {B} \/ (({C} \ {A}) \/ (({D,E} \/ {F,J,M}) \ {A})) by ENUMSET1:8 .= {B} \/ (({C} \ {A}) \/ (({D,E} \ {A}) \/ ({F,J,M} \ {A}))) by XBOOLE_1:42 .= {B} \/ (({C} \ {A}) \/ ({D,E} \/ ({F,J,M} \ {A}))) by A10, ZFMISC_1:63 .= {B} \/ (({C} \ {A}) \/ ({D,E} \/ (({F,J} \/ {M}) \ {A}))) by ENUMSET1:3 .= {B} \/ (({C} \ {A}) \/ ({D,E} \/ (({F,J} \ {A}) \/ ({M} \ {A})))) by XBOOLE_1:42 .= {B} \/ (({C} \ {A}) \/ ({D,E} \/ ({F,J} \/ ({M} \ {A})))) by A13, ZFMISC_1:63 .= {B} \/ ({C} \/ ({D,E} \/ ({F,J} \/ ({M} \ {A})))) by A11, ZFMISC_1:59 .= {B} \/ ({C} \/ ({D,E} \/ ({F,J} \/ {M}))) by A12, ZFMISC_1:59 .= {B} \/ ({C} \/ ({D,E} \/ {F,J,M})) by ENUMSET1:3 .= {B} \/ ({C} \/ {D,E,F,J,M}) by ENUMSET1:8 .= {B} \/ {C,D,E,F,J,M} by ENUMSET1:11 .= {B,C,D,E,F,J,M} by ENUMSET1:16 ; then A14: G \ {A} = {} \/ {B,C,D,E,F,J,M} by A9, XBOOLE_1:37; A15: '/\' (G \ {A}) c= (((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in '/\' (G \ {A}) or x in (((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M ) assume x in '/\' (G \ {A}) ; ::_thesis: x in (((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M then consider h being Function, FF being Subset-Family of Y such that A16: dom h = G \ {A} and A17: rng h = FF and A18: for d being set st d in G \ {A} holds h . d in d and A19: x = Intersect FF and A20: x <> {} by BVFUNC_2:def_1; A21: C in G \ {A} by A14, ENUMSET1:def_5; then A22: h . C in C by A18; set mbcdef = ((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F); set mbcde = (((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E); set mbcdefj = (((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F)) /\ (h . J); A23: not x in {{}} by A20, TARSKI:def_1; A24: J in G \ {A} by A14, ENUMSET1:def_5; then A25: h . J in rng h by A16, FUNCT_1:def_3; set mbc = (h . B) /\ (h . C); A26: B in G \ {A} by A14, ENUMSET1:def_5; then h . B in B by A18; then A27: (h . B) /\ (h . C) in INTERSECTION (B,C) by A22, SETFAM_1:def_5; A28: h . B in rng h by A16, A26, FUNCT_1:def_3; then A29: Intersect FF = meet (rng h) by A17, SETFAM_1:def_9; A30: h . C in rng h by A16, A21, FUNCT_1:def_3; A31: F in G \ {A} by A14, ENUMSET1:def_5; then A32: h . F in rng h by A16, FUNCT_1:def_3; set mbcd = ((h . B) /\ (h . C)) /\ (h . D); A33: E in G \ {A} by A14, ENUMSET1:def_5; then A34: h . E in rng h by A16, FUNCT_1:def_3; A35: M in G \ {A} by A14, ENUMSET1:def_5; then A36: h . M in rng h by A16, FUNCT_1:def_3; A37: D in G \ {A} by A14, ENUMSET1:def_5; then A38: h . D in rng h by A16, FUNCT_1:def_3; A39: x c= ((((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F)) /\ (h . J)) /\ (h . M) proof let p be set ; :: according to TARSKI:def_3 ::_thesis: ( not p in x or p in ((((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F)) /\ (h . J)) /\ (h . M) ) assume A40: p in x ; ::_thesis: p in ((((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F)) /\ (h . J)) /\ (h . M) then ( p in h . B & p in h . C ) by A19, A28, A30, A29, SETFAM_1:def_1; then A41: p in (h . B) /\ (h . C) by XBOOLE_0:def_4; p in h . D by A19, A38, A29, A40, SETFAM_1:def_1; then A42: p in ((h . B) /\ (h . C)) /\ (h . D) by A41, XBOOLE_0:def_4; p in h . E by A19, A34, A29, A40, SETFAM_1:def_1; then A43: p in (((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E) by A42, XBOOLE_0:def_4; p in h . F by A19, A32, A29, A40, SETFAM_1:def_1; then A44: p in ((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F) by A43, XBOOLE_0:def_4; p in h . J by A19, A25, A29, A40, SETFAM_1:def_1; then A45: p in (((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F)) /\ (h . J) by A44, XBOOLE_0:def_4; p in h . M by A19, A36, A29, A40, SETFAM_1:def_1; hence p in ((((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F)) /\ (h . J)) /\ (h . M) by A45, XBOOLE_0:def_4; ::_thesis: verum end; then ((h . B) /\ (h . C)) /\ (h . D) <> {} by A20; then A46: not ((h . B) /\ (h . C)) /\ (h . D) in {{}} by TARSKI:def_1; ((((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F)) /\ (h . J)) /\ (h . M) c= x proof A47: rng h c= {(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M)} proof let u be set ; :: according to TARSKI:def_3 ::_thesis: ( not u in rng h or u in {(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M)} ) assume u in rng h ; ::_thesis: u in {(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M)} then consider x1 being set such that A48: x1 in dom h and A49: u = h . x1 by FUNCT_1:def_3; ( x1 = B or x1 = C or x1 = D or x1 = E or x1 = F or x1 = J or x1 = M ) by A14, A16, A48, ENUMSET1:def_5; hence u in {(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M)} by A49, ENUMSET1:def_5; ::_thesis: verum end; let p be set ; :: according to TARSKI:def_3 ::_thesis: ( not p in ((((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F)) /\ (h . J)) /\ (h . M) or p in x ) assume A50: p in ((((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F)) /\ (h . J)) /\ (h . M) ; ::_thesis: p in x then A51: p in h . M by XBOOLE_0:def_4; A52: p in (((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F)) /\ (h . J) by A50, XBOOLE_0:def_4; then A53: p in h . J by XBOOLE_0:def_4; A54: p in ((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F) by A52, XBOOLE_0:def_4; then A55: p in (((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E) by XBOOLE_0:def_4; then A56: p in h . E by XBOOLE_0:def_4; A57: p in ((h . B) /\ (h . C)) /\ (h . D) by A55, XBOOLE_0:def_4; then A58: p in h . D by XBOOLE_0:def_4; p in (h . B) /\ (h . C) by A57, XBOOLE_0:def_4; then A59: ( p in h . B & p in h . C ) by XBOOLE_0:def_4; p in h . F by A54, XBOOLE_0:def_4; then for y being set st y in rng h holds p in y by A59, A58, A56, A53, A51, A47, ENUMSET1:def_5; hence p in x by A19, A28, A29, SETFAM_1:def_1; ::_thesis: verum end; then A60: ((((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F)) /\ (h . J)) /\ (h . M) = x by A39, XBOOLE_0:def_10; (h . B) /\ (h . C) <> {} by A20, A39; then not (h . B) /\ (h . C) in {{}} by TARSKI:def_1; then (h . B) /\ (h . C) in (INTERSECTION (B,C)) \ {{}} by A27, XBOOLE_0:def_5; then A61: (h . B) /\ (h . C) in B '/\' C by PARTIT1:def_4; h . D in D by A18, A37; then ((h . B) /\ (h . C)) /\ (h . D) in INTERSECTION ((B '/\' C),D) by A61, SETFAM_1:def_5; then ((h . B) /\ (h . C)) /\ (h . D) in (INTERSECTION ((B '/\' C),D)) \ {{}} by A46, XBOOLE_0:def_5; then A62: ((h . B) /\ (h . C)) /\ (h . D) in (B '/\' C) '/\' D by PARTIT1:def_4; (((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E) <> {} by A20, A39; then A63: not (((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E) in {{}} by TARSKI:def_1; h . E in E by A18, A33; then (((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E) in INTERSECTION (((B '/\' C) '/\' D),E) by A62, SETFAM_1:def_5; then (((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E) in (INTERSECTION (((B '/\' C) '/\' D),E)) \ {{}} by A63, XBOOLE_0:def_5; then A64: (((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E) in ((B '/\' C) '/\' D) '/\' E by PARTIT1:def_4; ((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F) <> {} by A20, A39; then A65: not ((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F) in {{}} by TARSKI:def_1; h . F in F by A18, A31; then ((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F) in INTERSECTION ((((B '/\' C) '/\' D) '/\' E),F) by A64, SETFAM_1:def_5; then ((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F) in (INTERSECTION ((((B '/\' C) '/\' D) '/\' E),F)) \ {{}} by A65, XBOOLE_0:def_5; then A66: ((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F) in (((B '/\' C) '/\' D) '/\' E) '/\' F by PARTIT1:def_4; (((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F)) /\ (h . J) <> {} by A20, A39; then A67: not (((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F)) /\ (h . J) in {{}} by TARSKI:def_1; h . J in J by A18, A24; then (((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F)) /\ (h . J) in INTERSECTION (((((B '/\' C) '/\' D) '/\' E) '/\' F),J) by A66, SETFAM_1:def_5; then (((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F)) /\ (h . J) in (INTERSECTION (((((B '/\' C) '/\' D) '/\' E) '/\' F),J)) \ {{}} by A67, XBOOLE_0:def_5; then A68: (((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F)) /\ (h . J) in ((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J by PARTIT1:def_4; h . M in M by A18, A35; then x in INTERSECTION ((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J),M) by A60, A68, SETFAM_1:def_5; then x in (INTERSECTION ((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J),M)) \ {{}} by A23, XBOOLE_0:def_5; hence x in (((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M by PARTIT1:def_4; ::_thesis: verum end; A69: (((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M c= '/\' (G \ {A}) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M or x in '/\' (G \ {A}) ) assume A70: x in (((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M ; ::_thesis: x in '/\' (G \ {A}) then A71: x <> {} by EQREL_1:def_4; x in (INTERSECTION ((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J),M)) \ {{}} by A70, PARTIT1:def_4; then consider bcdefj, m being set such that A72: bcdefj in ((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J and A73: m in M and A74: x = bcdefj /\ m by SETFAM_1:def_5; bcdefj in (INTERSECTION (((((B '/\' C) '/\' D) '/\' E) '/\' F),J)) \ {{}} by A72, PARTIT1:def_4; then consider bcdef, j being set such that A75: bcdef in (((B '/\' C) '/\' D) '/\' E) '/\' F and A76: j in J and A77: bcdefj = bcdef /\ j by SETFAM_1:def_5; bcdef in (INTERSECTION ((((B '/\' C) '/\' D) '/\' E),F)) \ {{}} by A75, PARTIT1:def_4; then consider bcde, f being set such that A78: bcde in ((B '/\' C) '/\' D) '/\' E and A79: f in F and A80: bcdef = bcde /\ f by SETFAM_1:def_5; bcde in (INTERSECTION (((B '/\' C) '/\' D),E)) \ {{}} by A78, PARTIT1:def_4; then consider bcd, e being set such that A81: bcd in (B '/\' C) '/\' D and A82: e in E and A83: bcde = bcd /\ e by SETFAM_1:def_5; bcd in (INTERSECTION ((B '/\' C),D)) \ {{}} by A81, PARTIT1:def_4; then consider bc, d being set such that A84: bc in B '/\' C and A85: d in D and A86: bcd = bc /\ d by SETFAM_1:def_5; bc in (INTERSECTION (B,C)) \ {{}} by A84, PARTIT1:def_4; then consider b, c being set such that A87: ( b in B & c in C ) and A88: bc = b /\ c by SETFAM_1:def_5; set h = ((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m); A89: (((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . B = b by A7, Th49; A90: dom (((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) = {M,B,C,D,E,F,J} by Th50 .= {M} \/ {B,C,D,E,F,J} by ENUMSET1:16 .= {B,C,D,E,F,J,M} by ENUMSET1:21 ; then A91: ( E in dom (((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) & F in dom (((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) ) by ENUMSET1:def_5; A92: D in dom (((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) by A90, ENUMSET1:def_5; then A93: (((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . D in rng (((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) by FUNCT_1:def_3; A94: ( J in dom (((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) & M in dom (((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) ) by A90, ENUMSET1:def_5; A95: ( B in dom (((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) & C in dom (((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) ) by A90, ENUMSET1:def_5; A96: {((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . B),((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . C),((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . D),((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . E),((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . F),((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . J),((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . M)} c= rng (((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) proof let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in {((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . B),((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . C),((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . D),((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . E),((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . F),((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . J),((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . M)} or t in rng (((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) ) assume t in {((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . B),((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . C),((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . D),((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . E),((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . F),((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . J),((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . M)} ; ::_thesis: t in rng (((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) then ( t = (((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . D or t = (((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . B or t = (((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . C or t = (((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . E or t = (((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . F or t = (((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . J or t = (((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . M ) by ENUMSET1:def_5; hence t in rng (((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) by A92, A95, A91, A94, FUNCT_1:def_3; ::_thesis: verum end; A97: for p being set st p in G \ {A} holds (((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . p in p proof let p be set ; ::_thesis: ( p in G \ {A} implies (((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . p in p ) assume p in G \ {A} ; ::_thesis: (((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . p in p then ( p = D or p = B or p = C or p = E or p = F or p = J or p = M ) by A14, ENUMSET1:def_5; hence (((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . p in p by A7, A73, A76, A79, A82, A85, A87, Th49; ::_thesis: verum end; A98: (((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . C = c by A7, Th49; A99: (((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . M = m by A7, Th49; A100: (((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . J = j by A7, Th49; A101: (((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . F = f by A7, Th49; A102: rng (((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) c= {((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . B),((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . C),((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . D),((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . E),((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . F),((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . J),((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . M)} proof let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in rng (((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) or t in {((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . B),((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . C),((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . D),((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . E),((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . F),((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . J),((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . M)} ) assume t in rng (((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) ; ::_thesis: t in {((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . B),((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . C),((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . D),((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . E),((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . F),((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . J),((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . M)} then consider x1 being set such that A103: x1 in dom (((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) and A104: t = (((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . x1 by FUNCT_1:def_3; ( x1 = D or x1 = B or x1 = C or x1 = E or x1 = F or x1 = J or x1 = M ) by A90, A103, ENUMSET1:def_5; hence t in {((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . B),((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . C),((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . D),((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . E),((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . F),((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . J),((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . M)} by A104, ENUMSET1:def_5; ::_thesis: verum end; then A105: rng (((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) = {((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . B),((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . C),((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . D),((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . E),((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . F),((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . J),((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . M)} by A96, XBOOLE_0:def_10; A106: (((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . E = e by A7, Th49; A107: (((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . D = d by A7, Th49; rng (((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) c= bool Y proof let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in rng (((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) or t in bool Y ) assume t in rng (((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) ; ::_thesis: t in bool Y then ( t = (((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . D or t = (((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . B or t = (((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . C or t = (((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . E or t = (((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . F or t = (((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . J or t = (((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) . M ) by A102, ENUMSET1:def_5; hence t in bool Y by A73, A76, A79, A82, A85, A87, A107, A89, A98, A106, A101, A100, A99; ::_thesis: verum end; then reconsider FF = rng (((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) as Subset-Family of Y ; reconsider h = ((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m) as Function ; A108: x c= Intersect FF proof let u be set ; :: according to TARSKI:def_3 ::_thesis: ( not u in x or u in Intersect FF ) assume A109: u in x ; ::_thesis: u in Intersect FF for y being set st y in FF holds u in y proof let y be set ; ::_thesis: ( y in FF implies u in y ) assume A110: y in FF ; ::_thesis: u in y now__::_thesis:_(_(_y_=_h_._D_&_u_in_y_)_or_(_y_=_h_._B_&_u_in_y_)_or_(_y_=_h_._C_&_u_in_y_)_or_(_y_=_h_._E_&_u_in_y_)_or_(_y_=_h_._F_&_u_in_y_)_or_(_y_=_h_._J_&_u_in_y_)_or_(_y_=_h_._M_&_u_in_y_)_) percases ( y = h . D or y = h . B or y = h . C or y = h . E or y = h . F or y = h . J or y = h . M ) by A102, A110, ENUMSET1:def_5; caseA111: y = h . D ; ::_thesis: u in y u in (((d /\ ((b /\ c) /\ e)) /\ f) /\ j) /\ m by A74, A77, A80, A83, A86, A88, A109, XBOOLE_1:16; then u in ((d /\ (((b /\ c) /\ e) /\ f)) /\ j) /\ m by XBOOLE_1:16; then u in (d /\ ((((b /\ c) /\ e) /\ f) /\ j)) /\ m by XBOOLE_1:16; then u in d /\ (((((b /\ c) /\ e) /\ f) /\ j) /\ m) by XBOOLE_1:16; hence u in y by A107, A111, XBOOLE_0:def_4; ::_thesis: verum end; caseA112: y = h . B ; ::_thesis: u in y u in ((((c /\ (d /\ b)) /\ e) /\ f) /\ j) /\ m by A74, A77, A80, A83, A86, A88, A109, XBOOLE_1:16; then u in (((c /\ ((d /\ b) /\ e)) /\ f) /\ j) /\ m by XBOOLE_1:16; then u in (((c /\ ((d /\ e) /\ b)) /\ f) /\ j) /\ m by XBOOLE_1:16; then u in ((c /\ (((d /\ e) /\ b) /\ f)) /\ j) /\ m by XBOOLE_1:16; then u in (c /\ ((((d /\ e) /\ b) /\ f) /\ j)) /\ m by XBOOLE_1:16; then u in (c /\ (((d /\ e) /\ (f /\ b)) /\ j)) /\ m by XBOOLE_1:16; then u in (c /\ ((d /\ e) /\ ((f /\ b) /\ j))) /\ m by XBOOLE_1:16; then u in (c /\ ((d /\ e) /\ (f /\ (j /\ b)))) /\ m by XBOOLE_1:16; then u in ((c /\ (d /\ e)) /\ (f /\ (j /\ b))) /\ m by XBOOLE_1:16; then u in (((c /\ (d /\ e)) /\ f) /\ (j /\ b)) /\ m by XBOOLE_1:16; then u in ((((c /\ (d /\ e)) /\ f) /\ j) /\ b) /\ m by XBOOLE_1:16; then u in ((((c /\ (d /\ e)) /\ f) /\ j) /\ m) /\ b by XBOOLE_1:16; hence u in y by A89, A112, XBOOLE_0:def_4; ::_thesis: verum end; caseA113: y = h . C ; ::_thesis: u in y u in ((((c /\ (d /\ b)) /\ e) /\ f) /\ j) /\ m by A74, A77, A80, A83, A86, A88, A109, XBOOLE_1:16; then u in (((c /\ ((d /\ b) /\ e)) /\ f) /\ j) /\ m by XBOOLE_1:16; then u in (((c /\ ((d /\ e) /\ b)) /\ f) /\ j) /\ m by XBOOLE_1:16; then u in ((c /\ (((d /\ e) /\ b) /\ f)) /\ j) /\ m by XBOOLE_1:16; then u in (c /\ ((((d /\ e) /\ b) /\ f) /\ j)) /\ m by XBOOLE_1:16; then u in c /\ (((((d /\ e) /\ b) /\ f) /\ j) /\ m) by XBOOLE_1:16; hence u in y by A98, A113, XBOOLE_0:def_4; ::_thesis: verum end; caseA114: y = h . E ; ::_thesis: u in y u in ((((b /\ c) /\ d) /\ (f /\ e)) /\ j) /\ m by A74, A77, A80, A83, A86, A88, A109, XBOOLE_1:16; then u in (((b /\ c) /\ d) /\ ((f /\ e) /\ j)) /\ m by XBOOLE_1:16; then u in (((b /\ c) /\ d) /\ ((f /\ j) /\ e)) /\ m by XBOOLE_1:16; then u in ((((b /\ c) /\ d) /\ (f /\ j)) /\ e) /\ m by XBOOLE_1:16; then u in ((((b /\ c) /\ d) /\ (f /\ j)) /\ m) /\ e by XBOOLE_1:16; hence u in y by A106, A114, XBOOLE_0:def_4; ::_thesis: verum end; caseA115: y = h . F ; ::_thesis: u in y u in (((((b /\ c) /\ d) /\ e) /\ j) /\ f) /\ m by A74, A77, A80, A83, A86, A88, A109, XBOOLE_1:16; then u in (((((b /\ c) /\ d) /\ e) /\ j) /\ m) /\ f by XBOOLE_1:16; hence u in y by A101, A115, XBOOLE_0:def_4; ::_thesis: verum end; caseA116: y = h . J ; ::_thesis: u in y u in (((((b /\ c) /\ d) /\ e) /\ f) /\ m) /\ j by A74, A77, A80, A83, A86, A88, A109, XBOOLE_1:16; hence u in y by A100, A116, XBOOLE_0:def_4; ::_thesis: verum end; case y = h . M ; ::_thesis: u in y hence u in y by A74, A99, A109, XBOOLE_0:def_4; ::_thesis: verum end; end; end; hence u in y ; ::_thesis: verum end; then u in meet FF by A105, SETFAM_1:def_1; hence u in Intersect FF by A105, SETFAM_1:def_9; ::_thesis: verum end; A117: Intersect FF = meet (rng h) by A93, SETFAM_1:def_9; Intersect FF c= x proof let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in Intersect FF or t in x ) assume A118: t in Intersect FF ; ::_thesis: t in x h . C in {(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M)} by ENUMSET1:def_5; then A119: t in c by A98, A96, A117, A118, SETFAM_1:def_1; h . B in {(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M)} by ENUMSET1:def_5; then t in b by A89, A96, A117, A118, SETFAM_1:def_1; then A120: t in b /\ c by A119, XBOOLE_0:def_4; h . D in {(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M)} by ENUMSET1:def_5; then t in d by A107, A96, A117, A118, SETFAM_1:def_1; then A121: t in (b /\ c) /\ d by A120, XBOOLE_0:def_4; h . E in {(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M)} by ENUMSET1:def_5; then t in e by A106, A96, A117, A118, SETFAM_1:def_1; then A122: t in ((b /\ c) /\ d) /\ e by A121, XBOOLE_0:def_4; h . F in {(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M)} by ENUMSET1:def_5; then t in f by A101, A96, A117, A118, SETFAM_1:def_1; then A123: t in (((b /\ c) /\ d) /\ e) /\ f by A122, XBOOLE_0:def_4; h . J in {(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M)} by ENUMSET1:def_5; then t in j by A100, A96, A117, A118, SETFAM_1:def_1; then A124: t in ((((b /\ c) /\ d) /\ e) /\ f) /\ j by A123, XBOOLE_0:def_4; h . M in {(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M)} by ENUMSET1:def_5; then t in m by A99, A96, A117, A118, SETFAM_1:def_1; hence t in x by A74, A77, A80, A83, A86, A88, A124, XBOOLE_0:def_4; ::_thesis: verum end; then x = Intersect FF by A108, XBOOLE_0:def_10; hence x in '/\' (G \ {A}) by A14, A90, A97, A71, BVFUNC_2:def_1; ::_thesis: verum end; CompF (A,G) = '/\' (G \ {A}) by BVFUNC_2:def_7; hence CompF (A,G) = (((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M by A69, A15, XBOOLE_0:def_10; ::_thesis: verum end; theorem Th55: :: BVFUNC14:55 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M being a_partition of Y st G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M holds CompF (B,G) = (((((A '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M being a_partition of Y st G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M holds CompF (B,G) = (((((A '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C, D, E, F, J, M being a_partition of Y st G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M holds CompF (B,G) = (((((A '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M let A, B, C, D, E, F, J, M be a_partition of Y; ::_thesis: ( G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M implies CompF (B,G) = (((((A '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M ) {A,B,C,D,E,F,J,M} = {A,B} \/ {C,D,E,F,J,M} by ENUMSET1:23 .= {B,A,C,D,E,F,J,M} by ENUMSET1:23 ; hence ( G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M implies CompF (B,G) = (((((A '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M ) by Th54; ::_thesis: verum end; theorem Th56: :: BVFUNC14:56 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M being a_partition of Y st G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M holds CompF (C,G) = (((((A '/\' B) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M being a_partition of Y st G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M holds CompF (C,G) = (((((A '/\' B) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C, D, E, F, J, M being a_partition of Y st G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M holds CompF (C,G) = (((((A '/\' B) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M let A, B, C, D, E, F, J, M be a_partition of Y; ::_thesis: ( G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M implies CompF (C,G) = (((((A '/\' B) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M ) {A,B,C,D,E,F,J,M} = {A,B,C} \/ {D,E,F,J,M} by ENUMSET1:24 .= ({A} \/ {B,C}) \/ {D,E,F,J,M} by ENUMSET1:2 .= {A,C,B} \/ {D,E,F,J,M} by ENUMSET1:2 .= {A,C,B,D,E,F,J,M} by ENUMSET1:24 ; hence ( G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M implies CompF (C,G) = (((((A '/\' B) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M ) by Th55; ::_thesis: verum end; theorem Th57: :: BVFUNC14:57 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M being a_partition of Y st G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M holds CompF (D,G) = (((((A '/\' B) '/\' C) '/\' E) '/\' F) '/\' J) '/\' M proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M being a_partition of Y st G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M holds CompF (D,G) = (((((A '/\' B) '/\' C) '/\' E) '/\' F) '/\' J) '/\' M let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C, D, E, F, J, M being a_partition of Y st G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M holds CompF (D,G) = (((((A '/\' B) '/\' C) '/\' E) '/\' F) '/\' J) '/\' M let A, B, C, D, E, F, J, M be a_partition of Y; ::_thesis: ( G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M implies CompF (D,G) = (((((A '/\' B) '/\' C) '/\' E) '/\' F) '/\' J) '/\' M ) {A,B,C,D,E,F,J,M} = {A,B} \/ {C,D,E,F,J,M} by ENUMSET1:23 .= {A,B} \/ ({C,D} \/ {E,F,J,M}) by ENUMSET1:12 .= {A,B} \/ {D,C,E,F,J,M} by ENUMSET1:12 .= {A,B,D,C,E,F,J,M} by ENUMSET1:23 ; hence ( G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M implies CompF (D,G) = (((((A '/\' B) '/\' C) '/\' E) '/\' F) '/\' J) '/\' M ) by Th56; ::_thesis: verum end; theorem Th58: :: BVFUNC14:58 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M being a_partition of Y st G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M holds CompF (E,G) = (((((A '/\' B) '/\' C) '/\' D) '/\' F) '/\' J) '/\' M proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M being a_partition of Y st G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M holds CompF (E,G) = (((((A '/\' B) '/\' C) '/\' D) '/\' F) '/\' J) '/\' M let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C, D, E, F, J, M being a_partition of Y st G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M holds CompF (E,G) = (((((A '/\' B) '/\' C) '/\' D) '/\' F) '/\' J) '/\' M let A, B, C, D, E, F, J, M be a_partition of Y; ::_thesis: ( G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M implies CompF (E,G) = (((((A '/\' B) '/\' C) '/\' D) '/\' F) '/\' J) '/\' M ) {A,B,C,D,E,F,J,M} = {A,B,C} \/ {D,E,F,J,M} by ENUMSET1:24 .= {A,B,C} \/ ({D,E} \/ {F,J,M}) by ENUMSET1:8 .= {A,B,C} \/ {E,D,F,J,M} by ENUMSET1:8 .= {A,B,C,E,D,F,J,M} by ENUMSET1:24 ; hence ( G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M implies CompF (E,G) = (((((A '/\' B) '/\' C) '/\' D) '/\' F) '/\' J) '/\' M ) by Th57; ::_thesis: verum end; theorem Th59: :: BVFUNC14:59 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M being a_partition of Y st G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M holds CompF (F,G) = (((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' J) '/\' M proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M being a_partition of Y st G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M holds CompF (F,G) = (((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' J) '/\' M let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C, D, E, F, J, M being a_partition of Y st G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M holds CompF (F,G) = (((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' J) '/\' M let A, B, C, D, E, F, J, M be a_partition of Y; ::_thesis: ( G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M implies CompF (F,G) = (((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' J) '/\' M ) {A,B,C,D,E,F,J,M} = {A,B,C,D} \/ {E,F,J,M} by ENUMSET1:25 .= {A,B,C,D} \/ ({E,F} \/ {J,M}) by ENUMSET1:5 .= {A,B,C,D} \/ {F,E,J,M} by ENUMSET1:5 .= {A,B,C,D,F,E,J,M} by ENUMSET1:25 ; hence ( G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M implies CompF (F,G) = (((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' J) '/\' M ) by Th58; ::_thesis: verum end; theorem Th60: :: BVFUNC14:60 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M being a_partition of Y st G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M holds CompF (J,G) = (((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' F) '/\' M proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M being a_partition of Y st G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M holds CompF (J,G) = (((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' F) '/\' M let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C, D, E, F, J, M being a_partition of Y st G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M holds CompF (J,G) = (((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' F) '/\' M let A, B, C, D, E, F, J, M be a_partition of Y; ::_thesis: ( G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M implies CompF (J,G) = (((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' F) '/\' M ) {A,B,C,D,E,F,J,M} = {A,B,C,D,E} \/ {F,J,M} by ENUMSET1:26 .= {A,B,C,D,E} \/ ({J,F} \/ {M}) by ENUMSET1:3 .= {A,B,C,D,E} \/ {J,F,M} by ENUMSET1:3 .= {A,B,C,D,E,J,F,M} by ENUMSET1:26 ; hence ( G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M implies CompF (J,G) = (((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' F) '/\' M ) by Th59; ::_thesis: verum end; theorem :: BVFUNC14:61 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M being a_partition of Y st G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M holds CompF (M,G) = (((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' F) '/\' J proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M being a_partition of Y st G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M holds CompF (M,G) = (((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' F) '/\' J let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C, D, E, F, J, M being a_partition of Y st G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M holds CompF (M,G) = (((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' F) '/\' J let A, B, C, D, E, F, J, M be a_partition of Y; ::_thesis: ( G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M implies CompF (M,G) = (((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' F) '/\' J ) {A,B,C,D,E,F,J,M} = {A,B,C,D,E,F} \/ {J,M} by ENUMSET1:27 .= {A,B,C,D,E,F,M,J} by ENUMSET1:27 ; hence ( G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M implies CompF (M,G) = (((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' F) '/\' J ) by Th60; ::_thesis: verum end; theorem Th62: :: BVFUNC14:62 for A, B, C, D, E, F, J, M being set for h being Function for A9, B9, C9, D9, E9, F9, J9, M9 being set st A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M & h = (((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (A .--> A9) holds ( h . B = B9 & h . C = C9 & h . D = D9 & h . E = E9 & h . F = F9 & h . J = J9 ) proof let A, B, C, D, E, F, J, M be set ; ::_thesis: for h being Function for A9, B9, C9, D9, E9, F9, J9, M9 being set st A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M & h = (((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (A .--> A9) holds ( h . B = B9 & h . C = C9 & h . D = D9 & h . E = E9 & h . F = F9 & h . J = J9 ) let h be Function; ::_thesis: for A9, B9, C9, D9, E9, F9, J9, M9 being set st A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M & h = (((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (A .--> A9) holds ( h . B = B9 & h . C = C9 & h . D = D9 & h . E = E9 & h . F = F9 & h . J = J9 ) let A9, B9, C9, D9, E9, F9, J9, M9 be set ; ::_thesis: ( A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M & h = (((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (A .--> A9) implies ( h . B = B9 & h . C = C9 & h . D = D9 & h . E = E9 & h . F = F9 & h . J = J9 ) ) assume that A1: A <> B and A2: A <> C and A3: A <> D and A4: A <> E and A5: A <> F and A6: A <> J and A7: ( B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M ) and A8: h = (((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (A .--> A9) ; ::_thesis: ( h . B = B9 & h . C = C9 & h . D = D9 & h . E = E9 & h . F = F9 & h . J = J9 ) A9: dom (A .--> A9) = {A} by FUNCOP_1:13; then not C in dom (A .--> A9) by A2, TARSKI:def_1; then A10: h . C = (((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) . C by A8, FUNCT_4:11; not J in dom (A .--> A9) by A6, A9, TARSKI:def_1; then A11: h . J = (((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) . J by A8, FUNCT_4:11 .= J9 by A7, Th49 ; not F in dom (A .--> A9) by A5, A9, TARSKI:def_1; then A12: h . F = (((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) . F by A8, FUNCT_4:11 .= F9 by A7, Th49 ; not E in dom (A .--> A9) by A4, A9, TARSKI:def_1; then A13: h . E = (((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) . E by A8, FUNCT_4:11 .= E9 by A7, Th49 ; not D in dom (A .--> A9) by A3, A9, TARSKI:def_1; then A14: h . D = (((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) . D by A8, FUNCT_4:11 .= D9 by A7, Th49 ; not B in dom (A .--> A9) by A1, A9, TARSKI:def_1; then h . B = (((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) . B by A8, FUNCT_4:11 .= B9 by A7, Th49 ; hence ( h . B = B9 & h . C = C9 & h . D = D9 & h . E = E9 & h . F = F9 & h . J = J9 ) by A7, A10, A14, A13, A12, A11, Th49; ::_thesis: verum end; theorem Th63: :: BVFUNC14:63 for A, B, C, D, E, F, J, M being set for h being Function for A9, B9, C9, D9, E9, F9, J9, M9 being set st h = (((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (A .--> A9) holds dom h = {A,B,C,D,E,F,J,M} proof let A, B, C, D, E, F, J, M be set ; ::_thesis: for h being Function for A9, B9, C9, D9, E9, F9, J9, M9 being set st h = (((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (A .--> A9) holds dom h = {A,B,C,D,E,F,J,M} let h be Function; ::_thesis: for A9, B9, C9, D9, E9, F9, J9, M9 being set st h = (((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (A .--> A9) holds dom h = {A,B,C,D,E,F,J,M} let A9, B9, C9, D9, E9, F9, J9, M9 be set ; ::_thesis: ( h = (((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (A .--> A9) implies dom h = {A,B,C,D,E,F,J,M} ) assume A1: h = (((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (A .--> A9) ; ::_thesis: dom h = {A,B,C,D,E,F,J,M} A2: dom (A .--> A9) = {A} by FUNCOP_1:13; dom (((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) = {M,B,C,D,E,F,J} by Th50 .= {M} \/ {B,C,D,E,F,J} by ENUMSET1:16 .= {B,C,D,E,F,J,M} by ENUMSET1:21 ; then dom ((((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (A .--> A9)) = {B,C,D,E,F,J,M} \/ {A} by A2, FUNCT_4:def_1 .= {A,B,C,D,E,F,J,M} by ENUMSET1:22 ; hence dom h = {A,B,C,D,E,F,J,M} by A1; ::_thesis: verum end; theorem Th64: :: BVFUNC14:64 for A, B, C, D, E, F, J, M being set for h being Function for A9, B9, C9, D9, E9, F9, J9, M9 being set st h = (((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (A .--> A9) holds rng h = {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M)} proof let A, B, C, D, E, F, J, M be set ; ::_thesis: for h being Function for A9, B9, C9, D9, E9, F9, J9, M9 being set st h = (((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (A .--> A9) holds rng h = {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M)} let h be Function; ::_thesis: for A9, B9, C9, D9, E9, F9, J9, M9 being set st h = (((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (A .--> A9) holds rng h = {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M)} let A9, B9, C9, D9, E9, F9, J9, M9 be set ; ::_thesis: ( h = (((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (A .--> A9) implies rng h = {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M)} ) assume h = (((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (A .--> A9) ; ::_thesis: rng h = {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M)} then A1: dom h = {A,B,C,D,E,F,J,M} by Th63; then B in dom h by ENUMSET1:def_6; then A2: h . B in rng h by FUNCT_1:def_3; M in dom h by A1, ENUMSET1:def_6; then A3: h . M in rng h by FUNCT_1:def_3; J in dom h by A1, ENUMSET1:def_6; then A4: h . J in rng h by FUNCT_1:def_3; F in dom h by A1, ENUMSET1:def_6; then A5: h . F in rng h by FUNCT_1:def_3; E in dom h by A1, ENUMSET1:def_6; then A6: h . E in rng h by FUNCT_1:def_3; A7: rng h c= {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M)} proof let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in rng h or t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M)} ) assume t in rng h ; ::_thesis: t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M)} then consider x1 being set such that A8: x1 in dom h and A9: t = h . x1 by FUNCT_1:def_3; ( x1 = A or x1 = B or x1 = C or x1 = D or x1 = E or x1 = F or x1 = J or x1 = M ) by A1, A8, ENUMSET1:def_6; hence t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M)} by A9, ENUMSET1:def_6; ::_thesis: verum end; D in dom h by A1, ENUMSET1:def_6; then A10: h . D in rng h by FUNCT_1:def_3; C in dom h by A1, ENUMSET1:def_6; then A11: h . C in rng h by FUNCT_1:def_3; A in dom h by A1, ENUMSET1:def_6; then A12: h . A in rng h by FUNCT_1:def_3; {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M)} c= rng h proof let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M)} or t in rng h ) assume t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M)} ; ::_thesis: t in rng h hence t in rng h by A12, A2, A11, A10, A6, A5, A4, A3, ENUMSET1:def_6; ::_thesis: verum end; hence rng h = {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M)} by A7, XBOOLE_0:def_10; ::_thesis: verum end; theorem :: BVFUNC14:65 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M being a_partition of Y for z, u being Element of Y st G is independent & G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M holds (EqClass (u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M))) /\ (EqClass (z,A)) <> {} proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M being a_partition of Y for z, u being Element of Y st G is independent & G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M holds (EqClass (u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M))) /\ (EqClass (z,A)) <> {} let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C, D, E, F, J, M being a_partition of Y for z, u being Element of Y st G is independent & G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M holds (EqClass (u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M))) /\ (EqClass (z,A)) <> {} let A, B, C, D, E, F, J, M be a_partition of Y; ::_thesis: for z, u being Element of Y st G is independent & G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M holds (EqClass (u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M))) /\ (EqClass (z,A)) <> {} let z, u be Element of Y; ::_thesis: ( G is independent & G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M implies (EqClass (u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M))) /\ (EqClass (z,A)) <> {} ) assume that A1: G is independent and A2: G = {A,B,C,D,E,F,J,M} and A3: ( A <> B & A <> C & A <> D & A <> E & A <> F & A <> J ) and A4: A <> M and A5: ( B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M ) ; ::_thesis: (EqClass (u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M))) /\ (EqClass (z,A)) <> {} set h = (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A))); A6: ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . B = EqClass (u,B) by A3, A5, Th62; reconsider GG = EqClass (u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M)) as set ; reconsider I = EqClass (z,A) as set ; GG = (EqClass (u,(((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J))) /\ (EqClass (u,M)) by Th1; then GG = ((EqClass (u,((((B '/\' C) '/\' D) '/\' E) '/\' F))) /\ (EqClass (u,J))) /\ (EqClass (u,M)) by Th1; then GG = (((EqClass (u,(((B '/\' C) '/\' D) '/\' E))) /\ (EqClass (u,F))) /\ (EqClass (u,J))) /\ (EqClass (u,M)) by Th1; then GG = ((((EqClass (u,((B '/\' C) '/\' D))) /\ (EqClass (u,E))) /\ (EqClass (u,F))) /\ (EqClass (u,J))) /\ (EqClass (u,M)) by Th1; then GG = (((((EqClass (u,(B '/\' C))) /\ (EqClass (u,D))) /\ (EqClass (u,E))) /\ (EqClass (u,F))) /\ (EqClass (u,J))) /\ (EqClass (u,M)) by Th1; then A7: GG /\ I = (((((((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E))) /\ (EqClass (u,F))) /\ (EqClass (u,J))) /\ (EqClass (u,M))) /\ (EqClass (z,A)) by Th1; A8: ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . A = EqClass (z,A) by FUNCT_7:94; A9: ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . C = EqClass (u,C) by A3, A5, Th62; A10: ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . M = EqClass (u,M) by A4, Lm2; A11: ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . J = EqClass (u,J) by A3, A5, Th62; A12: ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . F = EqClass (u,F) by A3, A5, Th62; A13: ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . E = EqClass (u,E) by A3, A5, Th62; A14: ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . D = EqClass (u,D) by A3, A5, Th62; A15: rng ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) = {(((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . A),(((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . B),(((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . C),(((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . D),(((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . E),(((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . F),(((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . J),(((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . M)} by Th64; rng ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) c= bool Y proof let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in rng ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) or t in bool Y ) assume t in rng ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) ; ::_thesis: t in bool Y then ( t = ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . A or t = ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . B or t = ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . C or t = ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . D or t = ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . E or t = ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . F or t = ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . J or t = ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . M ) by A15, ENUMSET1:def_6; hence t in bool Y by A8, A6, A9, A14, A13, A12, A11, A10; ::_thesis: verum end; then reconsider FF = rng ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) as Subset-Family of Y ; A16: dom ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) = G by A2, Th63; then A in dom ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) by A2, ENUMSET1:def_6; then A17: ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . A in rng ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then A18: Intersect FF = meet (rng ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A))))) by SETFAM_1:def_9; for d being set st d in G holds ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . d in d proof let d be set ; ::_thesis: ( d in G implies ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . d in d ) assume d in G ; ::_thesis: ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . d in d then ( d = A or d = B or d = C or d = D or d = E or d = F or d = J or d = M ) by A2, ENUMSET1:def_6; hence ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . d in d by A8, A6, A9, A14, A13, A12, A11, A10; ::_thesis: verum end; then Intersect FF <> {} by A1, A16, BVFUNC_2:def_5; then consider m being set such that A19: m in Intersect FF by XBOOLE_0:def_1; C in dom ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) by A2, A16, ENUMSET1:def_6; then ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . C in rng ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then A20: m in EqClass (u,C) by A9, A18, A19, SETFAM_1:def_1; B in dom ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) by A2, A16, ENUMSET1:def_6; then ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . B in rng ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then m in EqClass (u,B) by A6, A18, A19, SETFAM_1:def_1; then A21: m in (EqClass (u,B)) /\ (EqClass (u,C)) by A20, XBOOLE_0:def_4; D in dom ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) by A2, A16, ENUMSET1:def_6; then ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . D in rng ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then m in EqClass (u,D) by A14, A18, A19, SETFAM_1:def_1; then A22: m in ((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D)) by A21, XBOOLE_0:def_4; E in dom ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) by A2, A16, ENUMSET1:def_6; then ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . E in rng ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then m in EqClass (u,E) by A13, A18, A19, SETFAM_1:def_1; then A23: m in (((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E)) by A22, XBOOLE_0:def_4; F in dom ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) by A2, A16, ENUMSET1:def_6; then ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . F in rng ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then m in EqClass (u,F) by A12, A18, A19, SETFAM_1:def_1; then A24: m in ((((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E))) /\ (EqClass (u,F)) by A23, XBOOLE_0:def_4; J in dom ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) by A2, A16, ENUMSET1:def_6; then ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . J in rng ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then m in EqClass (u,J) by A11, A18, A19, SETFAM_1:def_1; then A25: m in (((((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E))) /\ (EqClass (u,F))) /\ (EqClass (u,J)) by A24, XBOOLE_0:def_4; M in dom ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) by A2, A16, ENUMSET1:def_6; then ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . M in rng ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then m in EqClass (u,M) by A10, A18, A19, SETFAM_1:def_1; then A26: m in ((((((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E))) /\ (EqClass (u,F))) /\ (EqClass (u,J))) /\ (EqClass (u,M)) by A25, XBOOLE_0:def_4; m in EqClass (z,A) by A8, A17, A18, A19, SETFAM_1:def_1; hence (EqClass (u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M))) /\ (EqClass (z,A)) <> {} by A7, A26, XBOOLE_0:def_4; ::_thesis: verum end; theorem :: BVFUNC14:66 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M being a_partition of Y for z, u being Element of Y st G is independent & G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M & EqClass (z,(((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M)) = EqClass (u,(((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M)) holds EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G))) proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M being a_partition of Y for z, u being Element of Y st G is independent & G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M & EqClass (z,(((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M)) = EqClass (u,(((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M)) holds EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G))) let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C, D, E, F, J, M being a_partition of Y for z, u being Element of Y st G is independent & G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M & EqClass (z,(((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M)) = EqClass (u,(((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M)) holds EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G))) let A, B, C, D, E, F, J, M be a_partition of Y; ::_thesis: for z, u being Element of Y st G is independent & G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M & EqClass (z,(((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M)) = EqClass (u,(((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M)) holds EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G))) let z, u be Element of Y; ::_thesis: ( G is independent & G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M & EqClass (z,(((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M)) = EqClass (u,(((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M)) implies EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G))) ) assume that A1: G is independent and A2: G = {A,B,C,D,E,F,J,M} and A3: ( A <> B & A <> C & A <> D & A <> E & A <> F & A <> J ) and A4: A <> M and A5: ( B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M ) and A6: EqClass (z,(((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M)) = EqClass (u,(((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M)) ; ::_thesis: EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G))) set h = (((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A))); A7: ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . B = EqClass (u,B) by A3, A5, Th62; set HH = EqClass (z,(CompF (B,G))); set I = EqClass (z,A); set GG = EqClass (u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M)); A8: EqClass (u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M)) = EqClass (u,(CompF (A,G))) by A2, A3, A4, A5, Th54; EqClass (u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M)) = (EqClass (u,(((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J))) /\ (EqClass (u,M)) by Th1; then EqClass (u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M)) = ((EqClass (u,((((B '/\' C) '/\' D) '/\' E) '/\' F))) /\ (EqClass (u,J))) /\ (EqClass (u,M)) by Th1; then EqClass (u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M)) = (((EqClass (u,(((B '/\' C) '/\' D) '/\' E))) /\ (EqClass (u,F))) /\ (EqClass (u,J))) /\ (EqClass (u,M)) by Th1; then EqClass (u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M)) = ((((EqClass (u,((B '/\' C) '/\' D))) /\ (EqClass (u,E))) /\ (EqClass (u,F))) /\ (EqClass (u,J))) /\ (EqClass (u,M)) by Th1; then EqClass (u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M)) = (((((EqClass (u,(B '/\' C))) /\ (EqClass (u,D))) /\ (EqClass (u,E))) /\ (EqClass (u,F))) /\ (EqClass (u,J))) /\ (EqClass (u,M)) by Th1; then A9: (EqClass (u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M))) /\ (EqClass (z,A)) = (((((((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E))) /\ (EqClass (u,F))) /\ (EqClass (u,J))) /\ (EqClass (u,M))) /\ (EqClass (z,A)) by Th1; A10: ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . A = EqClass (z,A) by FUNCT_7:94; A11: ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . C = EqClass (u,C) by A3, A5, Th62; A12: ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . M = EqClass (u,M) by A4, Lm2; A13: ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . J = EqClass (u,J) by A3, A5, Th62; A14: ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . F = EqClass (u,F) by A3, A5, Th62; A15: ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . E = EqClass (u,E) by A3, A5, Th62; A16: ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . D = EqClass (u,D) by A3, A5, Th62; A17: rng ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) = {(((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . A),(((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . B),(((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . C),(((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . D),(((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . E),(((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . F),(((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . J),(((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . M)} by Th64; rng ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) c= bool Y proof let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in rng ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) or t in bool Y ) assume t in rng ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) ; ::_thesis: t in bool Y then ( t = ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . A or t = ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . B or t = ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . C or t = ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . D or t = ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . E or t = ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . F or t = ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . J or t = ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . M ) by A17, ENUMSET1:def_6; hence t in bool Y by A10, A7, A11, A16, A15, A14, A13, A12; ::_thesis: verum end; then reconsider FF = rng ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) as Subset-Family of Y ; A18: dom ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) = G by A2, Th63; then A in dom ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) by A2, ENUMSET1:def_6; then A19: ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . A in rng ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then A20: Intersect FF = meet (rng ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A))))) by SETFAM_1:def_9; for d being set st d in G holds ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . d in d proof let d be set ; ::_thesis: ( d in G implies ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . d in d ) assume d in G ; ::_thesis: ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . d in d then ( d = A or d = B or d = C or d = D or d = E or d = F or d = J or d = M ) by A2, ENUMSET1:def_6; hence ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . d in d by A10, A7, A11, A16, A15, A14, A13, A12; ::_thesis: verum end; then Intersect FF <> {} by A1, A18, BVFUNC_2:def_5; then consider m being set such that A21: m in Intersect FF by XBOOLE_0:def_1; C in dom ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) by A2, A18, ENUMSET1:def_6; then ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . C in rng ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then A22: m in EqClass (u,C) by A11, A20, A21, SETFAM_1:def_1; B in dom ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) by A2, A18, ENUMSET1:def_6; then ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . B in rng ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then m in EqClass (u,B) by A7, A20, A21, SETFAM_1:def_1; then A23: m in (EqClass (u,B)) /\ (EqClass (u,C)) by A22, XBOOLE_0:def_4; D in dom ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) by A2, A18, ENUMSET1:def_6; then ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . D in rng ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then m in EqClass (u,D) by A16, A20, A21, SETFAM_1:def_1; then A24: m in ((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D)) by A23, XBOOLE_0:def_4; E in dom ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) by A2, A18, ENUMSET1:def_6; then ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . E in rng ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then m in EqClass (u,E) by A15, A20, A21, SETFAM_1:def_1; then A25: m in (((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E)) by A24, XBOOLE_0:def_4; F in dom ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) by A2, A18, ENUMSET1:def_6; then ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . F in rng ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then m in EqClass (u,F) by A14, A20, A21, SETFAM_1:def_1; then A26: m in ((((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E))) /\ (EqClass (u,F)) by A25, XBOOLE_0:def_4; J in dom ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) by A2, A18, ENUMSET1:def_6; then ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . J in rng ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then m in EqClass (u,J) by A13, A20, A21, SETFAM_1:def_1; then A27: m in (((((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E))) /\ (EqClass (u,F))) /\ (EqClass (u,J)) by A26, XBOOLE_0:def_4; M in dom ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) by A2, A18, ENUMSET1:def_6; then ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . M in rng ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then m in EqClass (u,M) by A12, A20, A21, SETFAM_1:def_1; then A28: m in ((((((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E))) /\ (EqClass (u,F))) /\ (EqClass (u,J))) /\ (EqClass (u,M)) by A27, XBOOLE_0:def_4; m in EqClass (z,A) by A10, A19, A20, A21, SETFAM_1:def_1; then (EqClass (u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M))) /\ (EqClass (z,A)) <> {} by A9, A28, XBOOLE_0:def_4; then consider p being set such that A29: p in (EqClass (u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M))) /\ (EqClass (z,A)) by XBOOLE_0:def_1; reconsider p = p as Element of Y by A29; reconsider K = EqClass (p,(((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M)) as set ; A30: p in EqClass (u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M)) by A29, XBOOLE_0:def_4; reconsider L = EqClass (z,(((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M)) as set ; A31: p in EqClass (p,(((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M)) by EQREL_1:def_6; EqClass (u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M)) = EqClass (u,(((((B '/\' (C '/\' D)) '/\' E) '/\' F) '/\' J) '/\' M)) by PARTIT1:14; then EqClass (u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M)) = EqClass (u,((((B '/\' ((C '/\' D) '/\' E)) '/\' F) '/\' J) '/\' M)) by PARTIT1:14; then EqClass (u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M)) = EqClass (u,(((B '/\' (((C '/\' D) '/\' E) '/\' F)) '/\' J) '/\' M)) by PARTIT1:14; then EqClass (u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M)) = EqClass (u,((B '/\' ((((C '/\' D) '/\' E) '/\' F) '/\' J)) '/\' M)) by PARTIT1:14; then EqClass (u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M)) = EqClass (u,(B '/\' (((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M))) by PARTIT1:14; then EqClass (u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M)) c= L by A6, BVFUNC11:3; then K meets L by A30, A31, XBOOLE_0:3; then K = L by EQREL_1:41; then A32: z in K by EQREL_1:def_6; A33: z in EqClass (z,(CompF (B,G))) by EQREL_1:def_6; z in EqClass (z,A) by EQREL_1:def_6; then z in (EqClass (z,A)) /\ K by A32, XBOOLE_0:def_4; then A34: (EqClass (z,A)) /\ K meets EqClass (z,(CompF (B,G))) by A33, XBOOLE_0:3; A35: A '/\' (((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) = (A '/\' ((((C '/\' D) '/\' E) '/\' F) '/\' J)) '/\' M by PARTIT1:14 .= ((A '/\' (((C '/\' D) '/\' E) '/\' F)) '/\' J) '/\' M by PARTIT1:14 .= (((A '/\' ((C '/\' D) '/\' E)) '/\' F) '/\' J) '/\' M by PARTIT1:14 .= ((((A '/\' (C '/\' D)) '/\' E) '/\' F) '/\' J) '/\' M by PARTIT1:14 .= (((((A '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M by PARTIT1:14 ; ( p in K & p in EqClass (z,A) ) by A29, EQREL_1:def_6, XBOOLE_0:def_4; then A36: p in (EqClass (z,A)) /\ K by XBOOLE_0:def_4; then ( (EqClass (z,A)) /\ K in INTERSECTION (A,(((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M)) & not (EqClass (z,A)) /\ K in {{}} ) by SETFAM_1:def_5, TARSKI:def_1; then (EqClass (z,A)) /\ K in (INTERSECTION (A,(((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M))) \ {{}} by XBOOLE_0:def_5; then A37: (EqClass (z,A)) /\ K in A '/\' (((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) by PARTIT1:def_4; CompF (B,G) = (((((A '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M by A2, A3, A4, A5, Th55; then p in EqClass (z,(CompF (B,G))) by A36, A37, A34, A35, EQREL_1:def_4; hence EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G))) by A8, A30, XBOOLE_0:3; ::_thesis: verum end; Lm5: for x1, x2, x3, x4, x5, x6, x7, x8, x9 being set holds {x1,x2,x3,x4,x5,x6,x7,x8,x9} = {x1,x2,x3,x4} \/ {x5,x6,x7,x8,x9} proof let x1, x2, x3, x4, x5, x6, x7, x8, x9 be set ; ::_thesis: {x1,x2,x3,x4,x5,x6,x7,x8,x9} = {x1,x2,x3,x4} \/ {x5,x6,x7,x8,x9} now__::_thesis:_for_x_being_set_holds_ (_x_in_{x1,x2,x3,x4,x5,x6,x7,x8,x9}_iff_x_in_{x1,x2,x3,x4}_\/_{x5,x6,x7,x8,x9}_) let x be set ; ::_thesis: ( x in {x1,x2,x3,x4,x5,x6,x7,x8,x9} iff x in {x1,x2,x3,x4} \/ {x5,x6,x7,x8,x9} ) A1: ( x in {x5,x6,x7,x8,x9} iff ( x = x5 or x = x6 or x = x7 or x = x8 or x = x9 ) ) by ENUMSET1:def_3; ( x in {x1,x2,x3,x4} iff ( x = x1 or x = x2 or x = x3 or x = x4 ) ) by ENUMSET1:def_2; hence ( x in {x1,x2,x3,x4,x5,x6,x7,x8,x9} iff x in {x1,x2,x3,x4} \/ {x5,x6,x7,x8,x9} ) by A1, ENUMSET1:def_7, XBOOLE_0:def_3; ::_thesis: verum end; hence {x1,x2,x3,x4,x5,x6,x7,x8,x9} = {x1,x2,x3,x4} \/ {x5,x6,x7,x8,x9} by TARSKI:1; ::_thesis: verum end; theorem Th67: :: BVFUNC14:67 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M, N being a_partition of Y st G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N holds CompF (A,G) = ((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M, N being a_partition of Y st G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N holds CompF (A,G) = ((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C, D, E, F, J, M, N being a_partition of Y st G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N holds CompF (A,G) = ((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N let A, B, C, D, E, F, J, M, N be a_partition of Y; ::_thesis: ( G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N implies CompF (A,G) = ((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N ) assume that A1: G = {A,B,C,D,E,F,J,M,N} and A2: A <> B and A3: A <> C and A4: ( A <> D & A <> E ) and A5: ( A <> F & A <> J ) and A6: ( A <> M & A <> N ) and A7: ( B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N ) and A8: M <> N ; ::_thesis: CompF (A,G) = ((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N A9: not B in {A} by A2, TARSKI:def_1; ( not D in {A} & not E in {A} ) by A4, TARSKI:def_1; then A10: {D,E} \ {A} = {D,E} by ZFMISC_1:63; A11: ( not F in {A} & not J in {A} ) by A5, TARSKI:def_1; A12: not C in {A} by A3, TARSKI:def_1; G \ {A} = ({A} \/ {B,C,D,E,F,J,M,N}) \ {A} by A1, ENUMSET1:77; then A13: G \ {A} = ({A} \ {A}) \/ ({B,C,D,E,F,J,M,N} \ {A}) by XBOOLE_1:42; A14: ( not M in {A} & not N in {A} ) by A6, TARSKI:def_1; {B,C,D,E,F,J,M,N} \ {A} = ({B} \/ {C,D,E,F,J,M,N}) \ {A} by ENUMSET1:22 .= ({B} \ {A}) \/ ({C,D,E,F,J,M,N} \ {A}) by XBOOLE_1:42 .= {B} \/ ({C,D,E,F,J,M,N} \ {A}) by A9, ZFMISC_1:59 .= {B} \/ (({C} \/ {D,E,F,J,M,N}) \ {A}) by ENUMSET1:16 .= {B} \/ (({C} \ {A}) \/ ({D,E,F,J,M,N} \ {A})) by XBOOLE_1:42 .= {B} \/ (({C} \ {A}) \/ (({D,E} \/ {F,J,M,N}) \ {A})) by ENUMSET1:12 .= {B} \/ (({C} \ {A}) \/ (({D,E} \ {A}) \/ ({F,J,M,N} \ {A}))) by XBOOLE_1:42 .= {B} \/ (({C} \ {A}) \/ ({D,E} \/ (({F,J} \/ {M,N}) \ {A}))) by A10, ENUMSET1:5 .= {B} \/ (({C} \ {A}) \/ ({D,E} \/ (({F,J} \ {A}) \/ ({M,N} \ {A})))) by XBOOLE_1:42 .= {B} \/ (({C} \ {A}) \/ ({D,E} \/ ({F,J} \/ ({M,N} \ {A})))) by A11, ZFMISC_1:63 .= {B} \/ ({C} \/ ({D,E} \/ ({F,J} \/ ({M,N} \ {A})))) by A12, ZFMISC_1:59 .= {B} \/ ({C} \/ ({D,E} \/ ({F,J} \/ {M,N}))) by A14, ZFMISC_1:63 .= {B} \/ ({C} \/ ({D,E} \/ {F,J,M,N})) by ENUMSET1:5 .= {B} \/ ({C} \/ {D,E,F,J,M,N}) by ENUMSET1:12 .= {B} \/ {C,D,E,F,J,M,N} by ENUMSET1:16 .= {B,C,D,E,F,J,M,N} by ENUMSET1:22 ; then A15: G \ {A} = {} \/ {B,C,D,E,F,J,M,N} by A13, XBOOLE_1:37; A16: '/\' (G \ {A}) c= ((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in '/\' (G \ {A}) or x in ((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N ) assume x in '/\' (G \ {A}) ; ::_thesis: x in ((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N then consider h being Function, FF being Subset-Family of Y such that A17: dom h = G \ {A} and A18: rng h = FF and A19: for d being set st d in G \ {A} holds h . d in d and A20: x = Intersect FF and A21: x <> {} by BVFUNC_2:def_1; A22: C in G \ {A} by A15, ENUMSET1:def_6; then A23: h . C in C by A19; set mbcd = ((h . B) /\ (h . C)) /\ (h . D); A24: E in G \ {A} by A15, ENUMSET1:def_6; then A25: h . E in rng h by A17, FUNCT_1:def_3; A26: N in G \ {A} by A15, ENUMSET1:def_6; then A27: h . N in rng h by A17, FUNCT_1:def_3; set mbc = (h . B) /\ (h . C); A28: B in G \ {A} by A15, ENUMSET1:def_6; then h . B in B by A19; then A29: (h . B) /\ (h . C) in INTERSECTION (B,C) by A23, SETFAM_1:def_5; A30: h . B in rng h by A17, A28, FUNCT_1:def_3; then A31: Intersect FF = meet (rng h) by A18, SETFAM_1:def_9; A32: h . C in rng h by A17, A22, FUNCT_1:def_3; A33: F in G \ {A} by A15, ENUMSET1:def_6; then A34: h . F in rng h by A17, FUNCT_1:def_3; A35: M in G \ {A} by A15, ENUMSET1:def_6; then A36: h . M in rng h by A17, FUNCT_1:def_3; A37: J in G \ {A} by A15, ENUMSET1:def_6; then A38: h . J in rng h by A17, FUNCT_1:def_3; A39: D in G \ {A} by A15, ENUMSET1:def_6; then A40: h . D in rng h by A17, FUNCT_1:def_3; A41: x c= (((((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F)) /\ (h . J)) /\ (h . M)) /\ (h . N) proof let p be set ; :: according to TARSKI:def_3 ::_thesis: ( not p in x or p in (((((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F)) /\ (h . J)) /\ (h . M)) /\ (h . N) ) assume A42: p in x ; ::_thesis: p in (((((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F)) /\ (h . J)) /\ (h . M)) /\ (h . N) then ( p in h . B & p in h . C ) by A20, A30, A32, A31, SETFAM_1:def_1; then A43: p in (h . B) /\ (h . C) by XBOOLE_0:def_4; p in h . D by A20, A40, A31, A42, SETFAM_1:def_1; then A44: p in ((h . B) /\ (h . C)) /\ (h . D) by A43, XBOOLE_0:def_4; p in h . E by A20, A25, A31, A42, SETFAM_1:def_1; then A45: p in (((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E) by A44, XBOOLE_0:def_4; p in h . F by A20, A34, A31, A42, SETFAM_1:def_1; then A46: p in ((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F) by A45, XBOOLE_0:def_4; p in h . J by A20, A38, A31, A42, SETFAM_1:def_1; then A47: p in (((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F)) /\ (h . J) by A46, XBOOLE_0:def_4; p in h . M by A20, A36, A31, A42, SETFAM_1:def_1; then A48: p in ((((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F)) /\ (h . J)) /\ (h . M) by A47, XBOOLE_0:def_4; p in h . N by A20, A27, A31, A42, SETFAM_1:def_1; hence p in (((((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F)) /\ (h . J)) /\ (h . M)) /\ (h . N) by A48, XBOOLE_0:def_4; ::_thesis: verum end; then ((h . B) /\ (h . C)) /\ (h . D) <> {} by A21; then A49: not ((h . B) /\ (h . C)) /\ (h . D) in {{}} by TARSKI:def_1; (h . B) /\ (h . C) <> {} by A21, A41; then not (h . B) /\ (h . C) in {{}} by TARSKI:def_1; then (h . B) /\ (h . C) in (INTERSECTION (B,C)) \ {{}} by A29, XBOOLE_0:def_5; then A50: (h . B) /\ (h . C) in B '/\' C by PARTIT1:def_4; h . D in D by A19, A39; then ((h . B) /\ (h . C)) /\ (h . D) in INTERSECTION ((B '/\' C),D) by A50, SETFAM_1:def_5; then ((h . B) /\ (h . C)) /\ (h . D) in (INTERSECTION ((B '/\' C),D)) \ {{}} by A49, XBOOLE_0:def_5; then A51: ((h . B) /\ (h . C)) /\ (h . D) in (B '/\' C) '/\' D by PARTIT1:def_4; set mbcdefjm = ((((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F)) /\ (h . J)) /\ (h . M); set mbcdefj = (((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F)) /\ (h . J); A52: not x in {{}} by A21, TARSKI:def_1; set mbcdef = ((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F); set mbcde = (((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E); ((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F) <> {} by A21, A41; then A53: not ((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F) in {{}} by TARSKI:def_1; (((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E) <> {} by A21, A41; then A54: not (((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E) in {{}} by TARSKI:def_1; (((((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F)) /\ (h . J)) /\ (h . M)) /\ (h . N) c= x proof A55: rng h c= {(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M),(h . N)} proof let u be set ; :: according to TARSKI:def_3 ::_thesis: ( not u in rng h or u in {(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M),(h . N)} ) assume u in rng h ; ::_thesis: u in {(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M),(h . N)} then consider x1 being set such that A56: x1 in dom h and A57: u = h . x1 by FUNCT_1:def_3; ( x1 = B or x1 = C or x1 = D or x1 = E or x1 = F or x1 = J or x1 = M or x1 = N ) by A15, A17, A56, ENUMSET1:def_6; hence u in {(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M),(h . N)} by A57, ENUMSET1:def_6; ::_thesis: verum end; let p be set ; :: according to TARSKI:def_3 ::_thesis: ( not p in (((((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F)) /\ (h . J)) /\ (h . M)) /\ (h . N) or p in x ) assume A58: p in (((((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F)) /\ (h . J)) /\ (h . M)) /\ (h . N) ; ::_thesis: p in x then A59: p in h . N by XBOOLE_0:def_4; A60: p in ((((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F)) /\ (h . J)) /\ (h . M) by A58, XBOOLE_0:def_4; then A61: p in (((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F)) /\ (h . J) by XBOOLE_0:def_4; then A62: p in h . J by XBOOLE_0:def_4; A63: p in h . M by A60, XBOOLE_0:def_4; A64: p in ((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F) by A61, XBOOLE_0:def_4; then A65: p in (((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E) by XBOOLE_0:def_4; then A66: p in h . E by XBOOLE_0:def_4; A67: p in ((h . B) /\ (h . C)) /\ (h . D) by A65, XBOOLE_0:def_4; then A68: p in h . D by XBOOLE_0:def_4; p in (h . B) /\ (h . C) by A67, XBOOLE_0:def_4; then A69: ( p in h . B & p in h . C ) by XBOOLE_0:def_4; p in h . F by A64, XBOOLE_0:def_4; then for y being set st y in rng h holds p in y by A69, A68, A66, A62, A63, A59, A55, ENUMSET1:def_6; hence p in x by A20, A30, A31, SETFAM_1:def_1; ::_thesis: verum end; then A70: (((((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F)) /\ (h . J)) /\ (h . M)) /\ (h . N) = x by A41, XBOOLE_0:def_10; h . E in E by A19, A24; then (((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E) in INTERSECTION (((B '/\' C) '/\' D),E) by A51, SETFAM_1:def_5; then (((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E) in (INTERSECTION (((B '/\' C) '/\' D),E)) \ {{}} by A54, XBOOLE_0:def_5; then A71: (((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E) in ((B '/\' C) '/\' D) '/\' E by PARTIT1:def_4; h . F in F by A19, A33; then ((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F) in INTERSECTION ((((B '/\' C) '/\' D) '/\' E),F) by A71, SETFAM_1:def_5; then ((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F) in (INTERSECTION ((((B '/\' C) '/\' D) '/\' E),F)) \ {{}} by A53, XBOOLE_0:def_5; then A72: ((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F) in (((B '/\' C) '/\' D) '/\' E) '/\' F by PARTIT1:def_4; (((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F)) /\ (h . J) <> {} by A21, A41; then A73: not (((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F)) /\ (h . J) in {{}} by TARSKI:def_1; h . J in J by A19, A37; then (((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F)) /\ (h . J) in INTERSECTION (((((B '/\' C) '/\' D) '/\' E) '/\' F),J) by A72, SETFAM_1:def_5; then (((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F)) /\ (h . J) in (INTERSECTION (((((B '/\' C) '/\' D) '/\' E) '/\' F),J)) \ {{}} by A73, XBOOLE_0:def_5; then A74: (((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F)) /\ (h . J) in ((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J by PARTIT1:def_4; ((((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F)) /\ (h . J)) /\ (h . M) <> {} by A21, A41; then A75: not ((((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F)) /\ (h . J)) /\ (h . M) in {{}} by TARSKI:def_1; h . M in M by A19, A35; then ((((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F)) /\ (h . J)) /\ (h . M) in INTERSECTION ((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J),M) by A74, SETFAM_1:def_5; then ((((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F)) /\ (h . J)) /\ (h . M) in (INTERSECTION ((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J),M)) \ {{}} by A75, XBOOLE_0:def_5; then A76: ((((((h . B) /\ (h . C)) /\ (h . D)) /\ (h . E)) /\ (h . F)) /\ (h . J)) /\ (h . M) in (((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M by PARTIT1:def_4; h . N in N by A19, A26; then x in INTERSECTION (((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M),N) by A70, A76, SETFAM_1:def_5; then x in (INTERSECTION (((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M),N)) \ {{}} by A52, XBOOLE_0:def_5; hence x in ((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N by PARTIT1:def_4; ::_thesis: verum end; A77: ((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N c= '/\' (G \ {A}) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in ((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N or x in '/\' (G \ {A}) ) assume A78: x in ((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N ; ::_thesis: x in '/\' (G \ {A}) then A79: x <> {} by EQREL_1:def_4; x in (INTERSECTION (((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M),N)) \ {{}} by A78, PARTIT1:def_4; then consider bcdefjm, n being set such that A80: bcdefjm in (((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M and A81: n in N and A82: x = bcdefjm /\ n by SETFAM_1:def_5; bcdefjm in (INTERSECTION ((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J),M)) \ {{}} by A80, PARTIT1:def_4; then consider bcdefj, m being set such that A83: bcdefj in ((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J and A84: m in M and A85: bcdefjm = bcdefj /\ m by SETFAM_1:def_5; bcdefj in (INTERSECTION (((((B '/\' C) '/\' D) '/\' E) '/\' F),J)) \ {{}} by A83, PARTIT1:def_4; then consider bcdef, j being set such that A86: bcdef in (((B '/\' C) '/\' D) '/\' E) '/\' F and A87: j in J and A88: bcdefj = bcdef /\ j by SETFAM_1:def_5; bcdef in (INTERSECTION ((((B '/\' C) '/\' D) '/\' E),F)) \ {{}} by A86, PARTIT1:def_4; then consider bcde, f being set such that A89: bcde in ((B '/\' C) '/\' D) '/\' E and A90: f in F and A91: bcdef = bcde /\ f by SETFAM_1:def_5; bcde in (INTERSECTION (((B '/\' C) '/\' D),E)) \ {{}} by A89, PARTIT1:def_4; then consider bcd, e being set such that A92: bcd in (B '/\' C) '/\' D and A93: e in E and A94: bcde = bcd /\ e by SETFAM_1:def_5; bcd in (INTERSECTION ((B '/\' C),D)) \ {{}} by A92, PARTIT1:def_4; then consider bc, d being set such that A95: bc in B '/\' C and A96: d in D and A97: bcd = bc /\ d by SETFAM_1:def_5; bc in (INTERSECTION (B,C)) \ {{}} by A95, PARTIT1:def_4; then consider b, c being set such that A98: b in B and A99: c in C and A100: bc = b /\ c by SETFAM_1:def_5; set h = (((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n); A101: ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . N = n by FUNCT_7:94; A102: dom ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) = {N,B,C,D,E,F,J,M} by Th63 .= {N} \/ {B,C,D,E,F,J,M} by ENUMSET1:22 .= {B,C,D,E,F,J,M,N} by ENUMSET1:28 ; then A103: C in dom ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) by ENUMSET1:def_6; A104: for p being set st p in G \ {A} holds ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . p in p proof let p be set ; ::_thesis: ( p in G \ {A} implies ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . p in p ) assume p in G \ {A} ; ::_thesis: ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . p in p then ( p = B or p = C or p = D or p = E or p = F or p = J or p = M or p = N ) by A15, ENUMSET1:def_6; hence ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . p in p by A7, A8, A81, A84, A87, A90, A93, A96, A98, A99, Lm2, Th62, FUNCT_7:94; ::_thesis: verum end; A105: D in dom ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) by A102, ENUMSET1:def_6; then A106: ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . D in rng ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) by FUNCT_1:def_3; A107: N in dom ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) by A102, ENUMSET1:def_6; A108: M in dom ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) by A102, ENUMSET1:def_6; A109: J in dom ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) by A102, ENUMSET1:def_6; A110: F in dom ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) by A102, ENUMSET1:def_6; A111: ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . B = b by A7, Th62; A112: rng ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) c= {(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . B),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . C),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . D),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . E),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . F),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . J),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . M),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . N)} proof let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in rng ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) or t in {(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . B),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . C),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . D),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . E),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . F),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . J),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . M),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . N)} ) assume t in rng ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) ; ::_thesis: t in {(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . B),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . C),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . D),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . E),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . F),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . J),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . M),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . N)} then consider x1 being set such that A113: x1 in dom ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) and A114: t = ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . x1 by FUNCT_1:def_3; now__::_thesis:_(_(_x1_=_D_&_t_in_{(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._B),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._C),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._D),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._E),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._F),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._J),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._M),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._N)}_)_or_(_x1_=_B_&_t_in_{(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._B),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._C),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._D),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._E),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._F),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._J),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._M),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._N)}_)_or_(_x1_=_C_&_t_in_{(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._B),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._C),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._D),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._E),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._F),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._J),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._M),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._N)}_)_or_(_x1_=_E_&_t_in_{(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._B),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._C),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._D),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._E),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._F),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._J),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._M),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._N)}_)_or_(_x1_=_F_&_t_in_{(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._B),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._C),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._D),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._E),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._F),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._J),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._M),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._N)}_)_or_(_x1_=_J_&_t_in_{(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._B),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._C),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._D),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._E),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._F),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._J),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._M),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._N)}_)_or_(_x1_=_M_&_t_in_{(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._B),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._C),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._D),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._E),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._F),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._J),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._M),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._N)}_)_or_(_x1_=_N_&_t_in_{(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._B),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._C),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._D),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._E),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._F),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._J),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._M),(((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._N)}_)_) percases ( x1 = D or x1 = B or x1 = C or x1 = E or x1 = F or x1 = J or x1 = M or x1 = N ) by A102, A113, ENUMSET1:def_6; case x1 = D ; ::_thesis: t in {(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . B),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . C),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . D),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . E),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . F),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . J),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . M),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . N)} hence t in {(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . B),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . C),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . D),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . E),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . F),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . J),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . M),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . N)} by A114, ENUMSET1:def_6; ::_thesis: verum end; case x1 = B ; ::_thesis: t in {(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . B),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . C),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . D),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . E),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . F),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . J),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . M),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . N)} hence t in {(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . B),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . C),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . D),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . E),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . F),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . J),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . M),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . N)} by A114, ENUMSET1:def_6; ::_thesis: verum end; case x1 = C ; ::_thesis: t in {(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . B),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . C),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . D),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . E),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . F),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . J),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . M),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . N)} hence t in {(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . B),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . C),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . D),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . E),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . F),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . J),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . M),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . N)} by A114, ENUMSET1:def_6; ::_thesis: verum end; case x1 = E ; ::_thesis: t in {(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . B),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . C),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . D),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . E),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . F),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . J),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . M),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . N)} hence t in {(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . B),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . C),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . D),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . E),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . F),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . J),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . M),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . N)} by A114, ENUMSET1:def_6; ::_thesis: verum end; case x1 = F ; ::_thesis: t in {(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . B),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . C),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . D),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . E),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . F),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . J),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . M),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . N)} hence t in {(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . B),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . C),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . D),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . E),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . F),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . J),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . M),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . N)} by A114, ENUMSET1:def_6; ::_thesis: verum end; case x1 = J ; ::_thesis: t in {(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . B),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . C),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . D),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . E),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . F),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . J),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . M),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . N)} hence t in {(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . B),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . C),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . D),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . E),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . F),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . J),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . M),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . N)} by A114, ENUMSET1:def_6; ::_thesis: verum end; case x1 = M ; ::_thesis: t in {(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . B),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . C),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . D),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . E),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . F),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . J),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . M),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . N)} hence t in {(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . B),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . C),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . D),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . E),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . F),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . J),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . M),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . N)} by A114, ENUMSET1:def_6; ::_thesis: verum end; case x1 = N ; ::_thesis: t in {(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . B),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . C),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . D),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . E),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . F),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . J),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . M),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . N)} hence t in {(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . B),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . C),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . D),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . E),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . F),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . J),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . M),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . N)} by A114, ENUMSET1:def_6; ::_thesis: verum end; end; end; hence t in {(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . B),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . C),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . D),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . E),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . F),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . J),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . M),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . N)} ; ::_thesis: verum end; A115: ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . J = j by A7, Th62; A116: ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . F = f by A7, Th62; A117: ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . M = m by A8, Lm2; A118: ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . E = e by A7, Th62; A119: ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . C = c by A7, Th62; A120: ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . D = d by A7, Th62; rng ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) c= bool Y proof let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in rng ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) or t in bool Y ) assume A121: t in rng ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) ; ::_thesis: t in bool Y now__::_thesis:_(_(_t_=_((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._D_&_t_in_bool_Y_)_or_(_t_=_((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._B_&_t_in_bool_Y_)_or_(_t_=_((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._C_&_t_in_bool_Y_)_or_(_t_=_((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._E_&_t_in_bool_Y_)_or_(_t_=_((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._F_&_t_in_bool_Y_)_or_(_t_=_((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._J_&_t_in_bool_Y_)_or_(_t_=_((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._M_&_t_in_bool_Y_)_or_(_t_=_((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._N_&_t_in_bool_Y_)_) percases ( t = ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . D or t = ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . B or t = ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . C or t = ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . E or t = ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . F or t = ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . J or t = ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . M or t = ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . N ) by A112, A121, ENUMSET1:def_6; case t = ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . D ; ::_thesis: t in bool Y hence t in bool Y by A96, A120; ::_thesis: verum end; case t = ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . B ; ::_thesis: t in bool Y hence t in bool Y by A98, A111; ::_thesis: verum end; case t = ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . C ; ::_thesis: t in bool Y hence t in bool Y by A99, A119; ::_thesis: verum end; case t = ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . E ; ::_thesis: t in bool Y hence t in bool Y by A93, A118; ::_thesis: verum end; case t = ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . F ; ::_thesis: t in bool Y hence t in bool Y by A90, A116; ::_thesis: verum end; case t = ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . J ; ::_thesis: t in bool Y hence t in bool Y by A87, A115; ::_thesis: verum end; case t = ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . M ; ::_thesis: t in bool Y hence t in bool Y by A84, A117; ::_thesis: verum end; case t = ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . N ; ::_thesis: t in bool Y hence t in bool Y by A81, A101; ::_thesis: verum end; end; end; hence t in bool Y ; ::_thesis: verum end; then reconsider FF = rng ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) as Subset-Family of Y ; A122: E in dom ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) by A102, ENUMSET1:def_6; A123: B in dom ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) by A102, ENUMSET1:def_6; {(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . B),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . C),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . D),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . E),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . F),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . J),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . M),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . N)} c= rng ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) proof let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in {(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . B),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . C),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . D),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . E),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . F),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . J),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . M),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . N)} or t in rng ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) ) assume A124: t in {(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . B),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . C),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . D),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . E),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . F),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . J),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . M),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . N)} ; ::_thesis: t in rng ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) now__::_thesis:_(_(_t_=_((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._D_&_t_in_rng_((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_)_or_(_t_=_((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._B_&_t_in_rng_((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_)_or_(_t_=_((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._C_&_t_in_rng_((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_)_or_(_t_=_((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._E_&_t_in_rng_((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_)_or_(_t_=_((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._F_&_t_in_rng_((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_)_or_(_t_=_((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._J_&_t_in_rng_((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_)_or_(_t_=_((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._M_&_t_in_rng_((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_)_or_(_t_=_((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_._N_&_t_in_rng_((((((((B_.-->_b)_+*_(C_.-->_c))_+*_(D_.-->_d))_+*_(E_.-->_e))_+*_(F_.-->_f))_+*_(J_.-->_j))_+*_(M_.-->_m))_+*_(N_.-->_n))_)_) percases ( t = ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . D or t = ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . B or t = ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . C or t = ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . E or t = ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . F or t = ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . J or t = ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . M or t = ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . N ) by A124, ENUMSET1:def_6; case t = ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . D ; ::_thesis: t in rng ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) hence t in rng ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) by A105, FUNCT_1:def_3; ::_thesis: verum end; case t = ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . B ; ::_thesis: t in rng ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) hence t in rng ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) by A123, FUNCT_1:def_3; ::_thesis: verum end; case t = ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . C ; ::_thesis: t in rng ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) hence t in rng ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) by A103, FUNCT_1:def_3; ::_thesis: verum end; case t = ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . E ; ::_thesis: t in rng ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) hence t in rng ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) by A122, FUNCT_1:def_3; ::_thesis: verum end; case t = ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . F ; ::_thesis: t in rng ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) hence t in rng ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) by A110, FUNCT_1:def_3; ::_thesis: verum end; case t = ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . J ; ::_thesis: t in rng ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) hence t in rng ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) by A109, FUNCT_1:def_3; ::_thesis: verum end; case t = ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . M ; ::_thesis: t in rng ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) hence t in rng ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) by A108, FUNCT_1:def_3; ::_thesis: verum end; case t = ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . N ; ::_thesis: t in rng ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) hence t in rng ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) by A107, FUNCT_1:def_3; ::_thesis: verum end; end; end; hence t in rng ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) ; ::_thesis: verum end; then A125: rng ((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) = {(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . B),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . C),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . D),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . E),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . F),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . J),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . M),(((((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n)) . N)} by A112, XBOOLE_0:def_10; reconsider h = (((((((B .--> b) +* (C .--> c)) +* (D .--> d)) +* (E .--> e)) +* (F .--> f)) +* (J .--> j)) +* (M .--> m)) +* (N .--> n) as Function ; A126: x c= Intersect FF proof let u be set ; :: according to TARSKI:def_3 ::_thesis: ( not u in x or u in Intersect FF ) assume A127: u in x ; ::_thesis: u in Intersect FF for y being set st y in FF holds u in y proof let y be set ; ::_thesis: ( y in FF implies u in y ) assume A128: y in FF ; ::_thesis: u in y now__::_thesis:_(_(_y_=_h_._D_&_u_in_y_)_or_(_y_=_h_._B_&_u_in_y_)_or_(_y_=_h_._C_&_u_in_y_)_or_(_y_=_h_._E_&_u_in_y_)_or_(_y_=_h_._F_&_u_in_y_)_or_(_y_=_h_._J_&_u_in_y_)_or_(_y_=_h_._M_&_u_in_y_)_or_(_y_=_h_._N_&_u_in_y_)_) percases ( y = h . D or y = h . B or y = h . C or y = h . E or y = h . F or y = h . J or y = h . M or y = h . N ) by A112, A128, ENUMSET1:def_6; caseA129: y = h . D ; ::_thesis: u in y u in ((((d /\ ((b /\ c) /\ e)) /\ f) /\ j) /\ m) /\ n by A82, A85, A88, A91, A94, A97, A100, A127, XBOOLE_1:16; then u in (((d /\ (((b /\ c) /\ e) /\ f)) /\ j) /\ m) /\ n by XBOOLE_1:16; then u in ((d /\ ((((b /\ c) /\ e) /\ f) /\ j)) /\ m) /\ n by XBOOLE_1:16; then u in (d /\ (((((b /\ c) /\ e) /\ f) /\ j) /\ m)) /\ n by XBOOLE_1:16; then u in d /\ ((((((b /\ c) /\ e) /\ f) /\ j) /\ m) /\ n) by XBOOLE_1:16; hence u in y by A120, A129, XBOOLE_0:def_4; ::_thesis: verum end; caseA130: y = h . B ; ::_thesis: u in y u in (((((c /\ (d /\ b)) /\ e) /\ f) /\ j) /\ m) /\ n by A82, A85, A88, A91, A94, A97, A100, A127, XBOOLE_1:16; then u in ((((c /\ ((d /\ b) /\ e)) /\ f) /\ j) /\ m) /\ n by XBOOLE_1:16; then u in ((((c /\ ((d /\ e) /\ b)) /\ f) /\ j) /\ m) /\ n by XBOOLE_1:16; then u in (((c /\ (((d /\ e) /\ b) /\ f)) /\ j) /\ m) /\ n by XBOOLE_1:16; then u in ((c /\ ((((d /\ e) /\ b) /\ f) /\ j)) /\ m) /\ n by XBOOLE_1:16; then u in ((c /\ (((d /\ e) /\ (f /\ b)) /\ j)) /\ m) /\ n by XBOOLE_1:16; then u in ((c /\ ((d /\ e) /\ ((f /\ b) /\ j))) /\ m) /\ n by XBOOLE_1:16; then u in ((c /\ ((d /\ e) /\ (f /\ (j /\ b)))) /\ m) /\ n by XBOOLE_1:16; then u in (((c /\ (d /\ e)) /\ (f /\ (j /\ b))) /\ m) /\ n by XBOOLE_1:16; then u in ((((c /\ (d /\ e)) /\ f) /\ (j /\ b)) /\ m) /\ n by XBOOLE_1:16; then u in (((((c /\ (d /\ e)) /\ f) /\ j) /\ b) /\ m) /\ n by XBOOLE_1:16; then u in ((((c /\ (d /\ e)) /\ f) /\ j) /\ (m /\ b)) /\ n by XBOOLE_1:16; then u in (((c /\ (d /\ e)) /\ f) /\ j) /\ ((m /\ b) /\ n) by XBOOLE_1:16; then u in (((c /\ (d /\ e)) /\ f) /\ j) /\ (m /\ (b /\ n)) by XBOOLE_1:16; then u in ((((c /\ (d /\ e)) /\ f) /\ j) /\ m) /\ (n /\ b) by XBOOLE_1:16; then u in (((((c /\ (d /\ e)) /\ f) /\ j) /\ m) /\ n) /\ b by XBOOLE_1:16; hence u in y by A111, A130, XBOOLE_0:def_4; ::_thesis: verum end; caseA131: y = h . C ; ::_thesis: u in y u in (((((c /\ (d /\ b)) /\ e) /\ f) /\ j) /\ m) /\ n by A82, A85, A88, A91, A94, A97, A100, A127, XBOOLE_1:16; then u in ((((c /\ ((d /\ b) /\ e)) /\ f) /\ j) /\ m) /\ n by XBOOLE_1:16; then u in ((((c /\ ((d /\ e) /\ b)) /\ f) /\ j) /\ m) /\ n by XBOOLE_1:16; then u in (((c /\ (((d /\ e) /\ b) /\ f)) /\ j) /\ m) /\ n by XBOOLE_1:16; then u in ((c /\ ((((d /\ e) /\ b) /\ f) /\ j)) /\ m) /\ n by XBOOLE_1:16; then u in (c /\ (((((d /\ e) /\ b) /\ f) /\ j) /\ m)) /\ n by XBOOLE_1:16; then u in c /\ ((((((d /\ e) /\ b) /\ f) /\ j) /\ m) /\ n) by XBOOLE_1:16; hence u in y by A119, A131, XBOOLE_0:def_4; ::_thesis: verum end; caseA132: y = h . E ; ::_thesis: u in y u in (((((b /\ c) /\ d) /\ (f /\ e)) /\ j) /\ m) /\ n by A82, A85, A88, A91, A94, A97, A100, A127, XBOOLE_1:16; then u in ((((b /\ c) /\ d) /\ ((f /\ e) /\ j)) /\ m) /\ n by XBOOLE_1:16; then u in ((((b /\ c) /\ d) /\ ((f /\ j) /\ e)) /\ m) /\ n by XBOOLE_1:16; then u in (((((b /\ c) /\ d) /\ (f /\ j)) /\ e) /\ m) /\ n by XBOOLE_1:16; then u in ((((b /\ c) /\ d) /\ (f /\ j)) /\ (e /\ m)) /\ n by XBOOLE_1:16; then u in (((b /\ c) /\ d) /\ (f /\ j)) /\ ((m /\ e) /\ n) by XBOOLE_1:16; then u in (((b /\ c) /\ d) /\ (f /\ j)) /\ (m /\ (n /\ e)) by XBOOLE_1:16; then u in ((((b /\ c) /\ d) /\ (f /\ j)) /\ m) /\ (n /\ e) by XBOOLE_1:16; then u in (((((b /\ c) /\ d) /\ (f /\ j)) /\ m) /\ n) /\ e by XBOOLE_1:16; hence u in y by A118, A132, XBOOLE_0:def_4; ::_thesis: verum end; caseA133: y = h . F ; ::_thesis: u in y u in ((((((b /\ c) /\ d) /\ e) /\ j) /\ f) /\ m) /\ n by A82, A85, A88, A91, A94, A97, A100, A127, XBOOLE_1:16; then u in ((((((b /\ c) /\ d) /\ e) /\ j) /\ m) /\ f) /\ n by XBOOLE_1:16; then u in ((((((b /\ c) /\ d) /\ e) /\ j) /\ m) /\ n) /\ f by XBOOLE_1:16; hence u in y by A116, A133, XBOOLE_0:def_4; ::_thesis: verum end; caseA134: y = h . J ; ::_thesis: u in y u in ((((((b /\ c) /\ d) /\ e) /\ f) /\ m) /\ j) /\ n by A82, A85, A88, A91, A94, A97, A100, A127, XBOOLE_1:16; then u in ((((((b /\ c) /\ d) /\ e) /\ f) /\ m) /\ n) /\ j by XBOOLE_1:16; hence u in y by A115, A134, XBOOLE_0:def_4; ::_thesis: verum end; caseA135: y = h . M ; ::_thesis: u in y u in ((((((b /\ c) /\ d) /\ e) /\ f) /\ j) /\ n) /\ m by A82, A85, A88, A91, A94, A97, A100, A127, XBOOLE_1:16; hence u in y by A117, A135, XBOOLE_0:def_4; ::_thesis: verum end; case y = h . N ; ::_thesis: u in y hence u in y by A82, A101, A127, XBOOLE_0:def_4; ::_thesis: verum end; end; end; hence u in y ; ::_thesis: verum end; then u in meet FF by A125, SETFAM_1:def_1; hence u in Intersect FF by A125, SETFAM_1:def_9; ::_thesis: verum end; A136: Intersect FF = meet (rng h) by A106, SETFAM_1:def_9; Intersect FF c= x proof let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in Intersect FF or t in x ) assume A137: t in Intersect FF ; ::_thesis: t in x h . C in rng h by A125, ENUMSET1:def_6; then A138: t in c by A119, A136, A137, SETFAM_1:def_1; h . B in rng h by A125, ENUMSET1:def_6; then t in b by A111, A136, A137, SETFAM_1:def_1; then A139: t in b /\ c by A138, XBOOLE_0:def_4; h . D in rng h by A125, ENUMSET1:def_6; then t in d by A120, A136, A137, SETFAM_1:def_1; then A140: t in (b /\ c) /\ d by A139, XBOOLE_0:def_4; h . E in rng h by A125, ENUMSET1:def_6; then t in e by A118, A136, A137, SETFAM_1:def_1; then A141: t in ((b /\ c) /\ d) /\ e by A140, XBOOLE_0:def_4; h . F in rng h by A125, ENUMSET1:def_6; then t in f by A116, A136, A137, SETFAM_1:def_1; then A142: t in (((b /\ c) /\ d) /\ e) /\ f by A141, XBOOLE_0:def_4; h . J in rng h by A125, ENUMSET1:def_6; then t in j by A115, A136, A137, SETFAM_1:def_1; then A143: t in ((((b /\ c) /\ d) /\ e) /\ f) /\ j by A142, XBOOLE_0:def_4; h . M in rng h by A125, ENUMSET1:def_6; then t in m by A117, A136, A137, SETFAM_1:def_1; then A144: t in (((((b /\ c) /\ d) /\ e) /\ f) /\ j) /\ m by A143, XBOOLE_0:def_4; h . N in rng h by A125, ENUMSET1:def_6; then t in n by A101, A136, A137, SETFAM_1:def_1; hence t in x by A82, A85, A88, A91, A94, A97, A100, A144, XBOOLE_0:def_4; ::_thesis: verum end; then x = Intersect FF by A126, XBOOLE_0:def_10; hence x in '/\' (G \ {A}) by A15, A102, A104, A79, BVFUNC_2:def_1; ::_thesis: verum end; CompF (A,G) = '/\' (G \ {A}) by BVFUNC_2:def_7; hence CompF (A,G) = ((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N by A77, A16, XBOOLE_0:def_10; ::_thesis: verum end; theorem Th68: :: BVFUNC14:68 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M, N being a_partition of Y st G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N holds CompF (B,G) = ((((((A '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M, N being a_partition of Y st G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N holds CompF (B,G) = ((((((A '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C, D, E, F, J, M, N being a_partition of Y st G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N holds CompF (B,G) = ((((((A '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N let A, B, C, D, E, F, J, M, N be a_partition of Y; ::_thesis: ( G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N implies CompF (B,G) = ((((((A '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N ) {A,B,C,D,E,F,J,M,N} = {A,B} \/ {C,D,E,F,J,M,N} by ENUMSET1:78 .= {B,A,C,D,E,F,J,M,N} by ENUMSET1:78 ; hence ( G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N implies CompF (B,G) = ((((((A '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N ) by Th67; ::_thesis: verum end; theorem Th69: :: BVFUNC14:69 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M, N being a_partition of Y st G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N holds CompF (C,G) = ((((((A '/\' B) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M, N being a_partition of Y st G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N holds CompF (C,G) = ((((((A '/\' B) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C, D, E, F, J, M, N being a_partition of Y st G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N holds CompF (C,G) = ((((((A '/\' B) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N let A, B, C, D, E, F, J, M, N be a_partition of Y; ::_thesis: ( G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N implies CompF (C,G) = ((((((A '/\' B) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N ) {A,B,C,D,E,F,J,M,N} = {A,B,C} \/ {D,E,F,J,M,N} by ENUMSET1:79 .= ({A} \/ {B,C}) \/ {D,E,F,J,M,N} by ENUMSET1:2 .= {A,C,B} \/ {D,E,F,J,M,N} by ENUMSET1:2 .= {A,C,B,D,E,F,J,M,N} by ENUMSET1:79 ; hence ( G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N implies CompF (C,G) = ((((((A '/\' B) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N ) by Th68; ::_thesis: verum end; theorem Th70: :: BVFUNC14:70 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M, N being a_partition of Y st G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N holds CompF (D,G) = ((((((A '/\' B) '/\' C) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M, N being a_partition of Y st G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N holds CompF (D,G) = ((((((A '/\' B) '/\' C) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C, D, E, F, J, M, N being a_partition of Y st G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N holds CompF (D,G) = ((((((A '/\' B) '/\' C) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N let A, B, C, D, E, F, J, M, N be a_partition of Y; ::_thesis: ( G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N implies CompF (D,G) = ((((((A '/\' B) '/\' C) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N ) {A,B,C,D,E,F,J,M,N} = {A,B} \/ {C,D,E,F,J,M,N} by ENUMSET1:78 .= {A,B} \/ ({C,D} \/ {E,F,J,M,N}) by ENUMSET1:17 .= {A,B} \/ {D,C,E,F,J,M,N} by ENUMSET1:17 .= {A,B,D,C,E,F,J,M,N} by ENUMSET1:78 ; hence ( G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N implies CompF (D,G) = ((((((A '/\' B) '/\' C) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N ) by Th69; ::_thesis: verum end; theorem Th71: :: BVFUNC14:71 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M, N being a_partition of Y st G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N holds CompF (E,G) = ((((((A '/\' B) '/\' C) '/\' D) '/\' F) '/\' J) '/\' M) '/\' N proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M, N being a_partition of Y st G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N holds CompF (E,G) = ((((((A '/\' B) '/\' C) '/\' D) '/\' F) '/\' J) '/\' M) '/\' N let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C, D, E, F, J, M, N being a_partition of Y st G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N holds CompF (E,G) = ((((((A '/\' B) '/\' C) '/\' D) '/\' F) '/\' J) '/\' M) '/\' N let A, B, C, D, E, F, J, M, N be a_partition of Y; ::_thesis: ( G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N implies CompF (E,G) = ((((((A '/\' B) '/\' C) '/\' D) '/\' F) '/\' J) '/\' M) '/\' N ) {A,B,C,D,E,F,J,M,N} = {A,B,C} \/ {D,E,F,J,M,N} by ENUMSET1:79 .= {A,B,C} \/ ({D,E} \/ {F,J,M,N}) by ENUMSET1:12 .= {A,B,C} \/ {E,D,F,J,M,N} by ENUMSET1:12 .= {A,B,C,E,D,F,J,M,N} by ENUMSET1:79 ; hence ( G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N implies CompF (E,G) = ((((((A '/\' B) '/\' C) '/\' D) '/\' F) '/\' J) '/\' M) '/\' N ) by Th70; ::_thesis: verum end; theorem Th72: :: BVFUNC14:72 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M, N being a_partition of Y st G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N holds CompF (F,G) = ((((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' J) '/\' M) '/\' N proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M, N being a_partition of Y st G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N holds CompF (F,G) = ((((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' J) '/\' M) '/\' N let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C, D, E, F, J, M, N being a_partition of Y st G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N holds CompF (F,G) = ((((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' J) '/\' M) '/\' N let A, B, C, D, E, F, J, M, N be a_partition of Y; ::_thesis: ( G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N implies CompF (F,G) = ((((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' J) '/\' M) '/\' N ) {A,B,C,D,E,F,J,M,N} = {A,B,C,D} \/ {E,F,J,M,N} by Lm5 .= {A,B,C,D} \/ ({E,F} \/ {J,M,N}) by ENUMSET1:8 .= {A,B,C,D} \/ {F,E,J,M,N} by ENUMSET1:8 .= {A,B,C,D,F,E,J,M,N} by Lm5 ; hence ( G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N implies CompF (F,G) = ((((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' J) '/\' M) '/\' N ) by Th71; ::_thesis: verum end; theorem Th73: :: BVFUNC14:73 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M, N being a_partition of Y st G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N holds CompF (J,G) = ((((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' F) '/\' M) '/\' N proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M, N being a_partition of Y st G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N holds CompF (J,G) = ((((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' F) '/\' M) '/\' N let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C, D, E, F, J, M, N being a_partition of Y st G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N holds CompF (J,G) = ((((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' F) '/\' M) '/\' N let A, B, C, D, E, F, J, M, N be a_partition of Y; ::_thesis: ( G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N implies CompF (J,G) = ((((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' F) '/\' M) '/\' N ) {A,B,C,D,E,F,J,M,N} = {A,B,C,D,E} \/ {F,J,M,N} by ENUMSET1:81 .= {A,B,C,D,E} \/ ({J,F} \/ {M,N}) by ENUMSET1:5 .= {A,B,C,D,E} \/ {J,F,M,N} by ENUMSET1:5 .= {A,B,C,D,E,J,F,M,N} by ENUMSET1:81 ; hence ( G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N implies CompF (J,G) = ((((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' F) '/\' M) '/\' N ) by Th72; ::_thesis: verum end; theorem Th74: :: BVFUNC14:74 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M, N being a_partition of Y st G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N holds CompF (M,G) = ((((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' N proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M, N being a_partition of Y st G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N holds CompF (M,G) = ((((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' N let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C, D, E, F, J, M, N being a_partition of Y st G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N holds CompF (M,G) = ((((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' N let A, B, C, D, E, F, J, M, N be a_partition of Y; ::_thesis: ( G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N implies CompF (M,G) = ((((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' N ) {A,B,C,D,E,F,J,M,N} = {A,B,C,D,E,F} \/ {J,M,N} by ENUMSET1:82 .= {A,B,C,D,E,F} \/ ({J,M} \/ {N}) by ENUMSET1:3 .= {A,B,C,D,E,F} \/ {M,J,N} by ENUMSET1:3 .= {A,B,C,D,E,F,M,J,N} by ENUMSET1:82 ; hence ( G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N implies CompF (M,G) = ((((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' N ) by Th73; ::_thesis: verum end; theorem :: BVFUNC14:75 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M, N being a_partition of Y st G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N holds CompF (N,G) = ((((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M, N being a_partition of Y st G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N holds CompF (N,G) = ((((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C, D, E, F, J, M, N being a_partition of Y st G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N holds CompF (N,G) = ((((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M let A, B, C, D, E, F, J, M, N be a_partition of Y; ::_thesis: ( G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N implies CompF (N,G) = ((((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M ) {A,B,C,D,E,F,J,M,N} = {A,B,C,D,E,F,J} \/ {M,N} by ENUMSET1:83 .= {A,B,C,D,E,F,J,N,M} by ENUMSET1:83 ; hence ( G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N implies CompF (N,G) = ((((((A '/\' B) '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M ) by Th74; ::_thesis: verum end; theorem Th76: :: BVFUNC14:76 for A, B, C, D, E, F, J, M, N being set for h being Function for A9, B9, C9, D9, E9, F9, J9, M9, N9 being set st A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N & h = ((((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (N .--> N9)) +* (A .--> A9) holds ( h . A = A9 & h . B = B9 & h . C = C9 & h . D = D9 & h . E = E9 & h . F = F9 & h . J = J9 & h . M = M9 & h . N = N9 ) proof let A, B, C, D, E, F, J, M, N be set ; ::_thesis: for h being Function for A9, B9, C9, D9, E9, F9, J9, M9, N9 being set st A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N & h = ((((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (N .--> N9)) +* (A .--> A9) holds ( h . A = A9 & h . B = B9 & h . C = C9 & h . D = D9 & h . E = E9 & h . F = F9 & h . J = J9 & h . M = M9 & h . N = N9 ) let h be Function; ::_thesis: for A9, B9, C9, D9, E9, F9, J9, M9, N9 being set st A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N & h = ((((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (N .--> N9)) +* (A .--> A9) holds ( h . A = A9 & h . B = B9 & h . C = C9 & h . D = D9 & h . E = E9 & h . F = F9 & h . J = J9 & h . M = M9 & h . N = N9 ) let A9, B9, C9, D9, E9, F9, J9, M9, N9 be set ; ::_thesis: ( A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N & h = ((((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (N .--> N9)) +* (A .--> A9) implies ( h . A = A9 & h . B = B9 & h . C = C9 & h . D = D9 & h . E = E9 & h . F = F9 & h . J = J9 & h . M = M9 & h . N = N9 ) ) assume that A1: A <> B and A2: A <> C and A3: A <> D and A4: A <> E and A5: A <> F and A6: A <> J and A7: A <> M and A8: A <> N and A9: ( B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N ) and A10: M <> N and A11: h = ((((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (N .--> N9)) +* (A .--> A9) ; ::_thesis: ( h . A = A9 & h . B = B9 & h . C = C9 & h . D = D9 & h . E = E9 & h . F = F9 & h . J = J9 & h . M = M9 & h . N = N9 ) A12: dom (A .--> A9) = {A} by FUNCOP_1:13; then A in dom (A .--> A9) by TARSKI:def_1; then A13: h . A = (A .--> A9) . A by A11, FUNCT_4:13; not E in dom (A .--> A9) by A4, A12, TARSKI:def_1; then A14: h . E = ((((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (N .--> N9)) . E by A11, FUNCT_4:11 .= E9 by A9, Th62 ; not N in dom (A .--> A9) by A8, A12, TARSKI:def_1; then A15: h . N = ((((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (N .--> N9)) . N by A11, FUNCT_4:11 .= N9 by FUNCT_7:94 ; not D in dom (A .--> A9) by A3, A12, TARSKI:def_1; then A16: h . D = ((((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (N .--> N9)) . D by A11, FUNCT_4:11 .= D9 by A9, Th62 ; not C in dom (A .--> A9) by A2, A12, TARSKI:def_1; then A17: h . C = ((((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (N .--> N9)) . C by A11, FUNCT_4:11; not J in dom (A .--> A9) by A6, A12, TARSKI:def_1; then A18: h . J = ((((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (N .--> N9)) . J by A11, FUNCT_4:11 .= J9 by A9, Th62 ; not F in dom (A .--> A9) by A5, A12, TARSKI:def_1; then A19: h . F = ((((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (N .--> N9)) . F by A11, FUNCT_4:11 .= F9 by A9, Th62 ; not M in dom (A .--> A9) by A7, A12, TARSKI:def_1; then A20: h . M = ((((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (N .--> N9)) . M by A11, FUNCT_4:11 .= M9 by A10, Lm2 ; not B in dom (A .--> A9) by A1, A12, TARSKI:def_1; then h . B = ((((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (N .--> N9)) . B by A11, FUNCT_4:11 .= B9 by A9, Th62 ; hence ( h . A = A9 & h . B = B9 & h . C = C9 & h . D = D9 & h . E = E9 & h . F = F9 & h . J = J9 & h . M = M9 & h . N = N9 ) by A9, A13, A17, A16, A14, A19, A18, A20, A15, Th62, FUNCOP_1:72; ::_thesis: verum end; theorem Th77: :: BVFUNC14:77 for A, B, C, D, E, F, J, M, N being set for h being Function for A9, B9, C9, D9, E9, F9, J9, M9, N9 being set st h = ((((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (N .--> N9)) +* (A .--> A9) holds dom h = {A,B,C,D,E,F,J,M,N} proof let A, B, C, D, E, F, J, M, N be set ; ::_thesis: for h being Function for A9, B9, C9, D9, E9, F9, J9, M9, N9 being set st h = ((((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (N .--> N9)) +* (A .--> A9) holds dom h = {A,B,C,D,E,F,J,M,N} let h be Function; ::_thesis: for A9, B9, C9, D9, E9, F9, J9, M9, N9 being set st h = ((((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (N .--> N9)) +* (A .--> A9) holds dom h = {A,B,C,D,E,F,J,M,N} let A9, B9, C9, D9, E9, F9, J9, M9, N9 be set ; ::_thesis: ( h = ((((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (N .--> N9)) +* (A .--> A9) implies dom h = {A,B,C,D,E,F,J,M,N} ) assume A1: h = ((((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (N .--> N9)) +* (A .--> A9) ; ::_thesis: dom h = {A,B,C,D,E,F,J,M,N} A2: dom (A .--> A9) = {A} by FUNCOP_1:13; dom ((((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (N .--> N9)) = {N,B,C,D,E,F,J,M} by Th63 .= {N} \/ {B,C,D,E,F,J,M} by ENUMSET1:22 .= {B,C,D,E,F,J,M,N} by ENUMSET1:28 ; then dom (((((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (N .--> N9)) +* (A .--> A9)) = {B,C,D,E,F,J,M,N} \/ {A} by A2, FUNCT_4:def_1 .= {A,B,C,D,E,F,J,M,N} by ENUMSET1:77 ; hence dom h = {A,B,C,D,E,F,J,M,N} by A1; ::_thesis: verum end; theorem Th78: :: BVFUNC14:78 for A, B, C, D, E, F, J, M, N being set for h being Function for A9, B9, C9, D9, E9, F9, J9, M9, N9 being set st h = ((((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (N .--> N9)) +* (A .--> A9) holds rng h = {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M),(h . N)} proof let A, B, C, D, E, F, J, M, N be set ; ::_thesis: for h being Function for A9, B9, C9, D9, E9, F9, J9, M9, N9 being set st h = ((((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (N .--> N9)) +* (A .--> A9) holds rng h = {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M),(h . N)} let h be Function; ::_thesis: for A9, B9, C9, D9, E9, F9, J9, M9, N9 being set st h = ((((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (N .--> N9)) +* (A .--> A9) holds rng h = {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M),(h . N)} let A9, B9, C9, D9, E9, F9, J9, M9, N9 be set ; ::_thesis: ( h = ((((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (N .--> N9)) +* (A .--> A9) implies rng h = {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M),(h . N)} ) assume h = ((((((((B .--> B9) +* (C .--> C9)) +* (D .--> D9)) +* (E .--> E9)) +* (F .--> F9)) +* (J .--> J9)) +* (M .--> M9)) +* (N .--> N9)) +* (A .--> A9) ; ::_thesis: rng h = {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M),(h . N)} then A1: dom h = {A,B,C,D,E,F,J,M,N} by Th77; then A2: B in dom h by ENUMSET1:def_7; A3: M in dom h by A1, ENUMSET1:def_7; A4: J in dom h by A1, ENUMSET1:def_7; A5: N in dom h by A1, ENUMSET1:def_7; A6: D in dom h by A1, ENUMSET1:def_7; A7: C in dom h by A1, ENUMSET1:def_7; A8: rng h c= {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M),(h . N)} proof let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in rng h or t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M),(h . N)} ) assume t in rng h ; ::_thesis: t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M),(h . N)} then consider x1 being set such that A9: x1 in dom h and A10: t = h . x1 by FUNCT_1:def_3; now__::_thesis:_(_(_x1_=_A_&_t_in_{(h_._A),(h_._B),(h_._C),(h_._D),(h_._E),(h_._F),(h_._J),(h_._M),(h_._N)}_)_or_(_x1_=_B_&_t_in_{(h_._A),(h_._B),(h_._C),(h_._D),(h_._E),(h_._F),(h_._J),(h_._M),(h_._N)}_)_or_(_x1_=_C_&_t_in_{(h_._A),(h_._B),(h_._C),(h_._D),(h_._E),(h_._F),(h_._J),(h_._M),(h_._N)}_)_or_(_x1_=_D_&_t_in_{(h_._A),(h_._B),(h_._C),(h_._D),(h_._E),(h_._F),(h_._J),(h_._M),(h_._N)}_)_or_(_x1_=_E_&_t_in_{(h_._A),(h_._B),(h_._C),(h_._D),(h_._E),(h_._F),(h_._J),(h_._M),(h_._N)}_)_or_(_x1_=_F_&_t_in_{(h_._A),(h_._B),(h_._C),(h_._D),(h_._E),(h_._F),(h_._J),(h_._M),(h_._N)}_)_or_(_x1_=_J_&_t_in_{(h_._A),(h_._B),(h_._C),(h_._D),(h_._E),(h_._F),(h_._J),(h_._M),(h_._N)}_)_or_(_x1_=_M_&_t_in_{(h_._A),(h_._B),(h_._C),(h_._D),(h_._E),(h_._F),(h_._J),(h_._M),(h_._N)}_)_or_(_x1_=_N_&_t_in_{(h_._A),(h_._B),(h_._C),(h_._D),(h_._E),(h_._F),(h_._J),(h_._M),(h_._N)}_)_) percases ( x1 = A or x1 = B or x1 = C or x1 = D or x1 = E or x1 = F or x1 = J or x1 = M or x1 = N ) by A1, A9, ENUMSET1:def_7; case x1 = A ; ::_thesis: t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M),(h . N)} hence t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M),(h . N)} by A10, ENUMSET1:def_7; ::_thesis: verum end; case x1 = B ; ::_thesis: t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M),(h . N)} hence t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M),(h . N)} by A10, ENUMSET1:def_7; ::_thesis: verum end; case x1 = C ; ::_thesis: t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M),(h . N)} hence t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M),(h . N)} by A10, ENUMSET1:def_7; ::_thesis: verum end; case x1 = D ; ::_thesis: t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M),(h . N)} hence t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M),(h . N)} by A10, ENUMSET1:def_7; ::_thesis: verum end; case x1 = E ; ::_thesis: t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M),(h . N)} hence t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M),(h . N)} by A10, ENUMSET1:def_7; ::_thesis: verum end; case x1 = F ; ::_thesis: t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M),(h . N)} hence t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M),(h . N)} by A10, ENUMSET1:def_7; ::_thesis: verum end; case x1 = J ; ::_thesis: t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M),(h . N)} hence t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M),(h . N)} by A10, ENUMSET1:def_7; ::_thesis: verum end; case x1 = M ; ::_thesis: t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M),(h . N)} hence t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M),(h . N)} by A10, ENUMSET1:def_7; ::_thesis: verum end; case x1 = N ; ::_thesis: t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M),(h . N)} hence t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M),(h . N)} by A10, ENUMSET1:def_7; ::_thesis: verum end; end; end; hence t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M),(h . N)} ; ::_thesis: verum end; A11: F in dom h by A1, ENUMSET1:def_7; A12: E in dom h by A1, ENUMSET1:def_7; A13: A in dom h by A1, ENUMSET1:def_7; {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M),(h . N)} c= rng h proof let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M),(h . N)} or t in rng h ) assume A14: t in {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M),(h . N)} ; ::_thesis: t in rng h now__::_thesis:_(_(_t_=_h_._A_&_t_in_rng_h_)_or_(_t_=_h_._B_&_t_in_rng_h_)_or_(_t_=_h_._C_&_t_in_rng_h_)_or_(_t_=_h_._D_&_t_in_rng_h_)_or_(_t_=_h_._E_&_t_in_rng_h_)_or_(_t_=_h_._F_&_t_in_rng_h_)_or_(_t_=_h_._J_&_t_in_rng_h_)_or_(_t_=_h_._M_&_t_in_rng_h_)_or_(_t_=_h_._N_&_t_in_rng_h_)_) percases ( t = h . A or t = h . B or t = h . C or t = h . D or t = h . E or t = h . F or t = h . J or t = h . M or t = h . N ) by A14, ENUMSET1:def_7; case t = h . A ; ::_thesis: t in rng h hence t in rng h by A13, FUNCT_1:def_3; ::_thesis: verum end; case t = h . B ; ::_thesis: t in rng h hence t in rng h by A2, FUNCT_1:def_3; ::_thesis: verum end; case t = h . C ; ::_thesis: t in rng h hence t in rng h by A7, FUNCT_1:def_3; ::_thesis: verum end; case t = h . D ; ::_thesis: t in rng h hence t in rng h by A6, FUNCT_1:def_3; ::_thesis: verum end; case t = h . E ; ::_thesis: t in rng h hence t in rng h by A12, FUNCT_1:def_3; ::_thesis: verum end; case t = h . F ; ::_thesis: t in rng h hence t in rng h by A11, FUNCT_1:def_3; ::_thesis: verum end; case t = h . J ; ::_thesis: t in rng h hence t in rng h by A4, FUNCT_1:def_3; ::_thesis: verum end; case t = h . M ; ::_thesis: t in rng h hence t in rng h by A3, FUNCT_1:def_3; ::_thesis: verum end; case t = h . N ; ::_thesis: t in rng h hence t in rng h by A5, FUNCT_1:def_3; ::_thesis: verum end; end; end; hence t in rng h ; ::_thesis: verum end; hence rng h = {(h . A),(h . B),(h . C),(h . D),(h . E),(h . F),(h . J),(h . M),(h . N)} by A8, XBOOLE_0:def_10; ::_thesis: verum end; theorem :: BVFUNC14:79 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M, N being a_partition of Y for z, u being Element of Y st G is independent & G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N holds (EqClass (u,(((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N))) /\ (EqClass (z,A)) <> {} proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M, N being a_partition of Y for z, u being Element of Y st G is independent & G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N holds (EqClass (u,(((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N))) /\ (EqClass (z,A)) <> {} let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C, D, E, F, J, M, N being a_partition of Y for z, u being Element of Y st G is independent & G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N holds (EqClass (u,(((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N))) /\ (EqClass (z,A)) <> {} let A, B, C, D, E, F, J, M, N be a_partition of Y; ::_thesis: for z, u being Element of Y st G is independent & G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N holds (EqClass (u,(((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N))) /\ (EqClass (z,A)) <> {} let z, u be Element of Y; ::_thesis: ( G is independent & G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N implies (EqClass (u,(((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N))) /\ (EqClass (z,A)) <> {} ) assume that A1: G is independent and A2: G = {A,B,C,D,E,F,J,M,N} and A3: ( A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N ) ; ::_thesis: (EqClass (u,(((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N))) /\ (EqClass (z,A)) <> {} set h = ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A))); A4: (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . A = EqClass (z,A) by A3, Th76; set GG = EqClass (u,(((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)); EqClass (u,(((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)) = (EqClass (u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M))) /\ (EqClass (u,N)) by Th1; then EqClass (u,(((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)) = ((EqClass (u,(((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J))) /\ (EqClass (u,M))) /\ (EqClass (u,N)) by Th1; then EqClass (u,(((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)) = (((EqClass (u,((((B '/\' C) '/\' D) '/\' E) '/\' F))) /\ (EqClass (u,J))) /\ (EqClass (u,M))) /\ (EqClass (u,N)) by Th1; then EqClass (u,(((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)) = ((((EqClass (u,(((B '/\' C) '/\' D) '/\' E))) /\ (EqClass (u,F))) /\ (EqClass (u,J))) /\ (EqClass (u,M))) /\ (EqClass (u,N)) by Th1; then EqClass (u,(((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)) = (((((EqClass (u,((B '/\' C) '/\' D))) /\ (EqClass (u,E))) /\ (EqClass (u,F))) /\ (EqClass (u,J))) /\ (EqClass (u,M))) /\ (EqClass (u,N)) by Th1; then EqClass (u,(((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)) = ((((((EqClass (u,(B '/\' C))) /\ (EqClass (u,D))) /\ (EqClass (u,E))) /\ (EqClass (u,F))) /\ (EqClass (u,J))) /\ (EqClass (u,M))) /\ (EqClass (u,N)) by Th1; then A5: (EqClass (u,(((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N))) /\ (EqClass (z,A)) = ((((((((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E))) /\ (EqClass (u,F))) /\ (EqClass (u,J))) /\ (EqClass (u,M))) /\ (EqClass (u,N))) /\ (EqClass (z,A)) by Th1; A6: (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . B = EqClass (u,B) by A3, Th76; A7: (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . F = EqClass (u,F) by A3, Th76; A8: (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . E = EqClass (u,E) by A3, Th76; A9: (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . M = EqClass (u,M) by A3, Th76; A10: (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . J = EqClass (u,J) by A3, Th76; A11: (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . N = EqClass (u,N) by A3, Th76; A12: (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . D = EqClass (u,D) by A3, Th76; A13: (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . C = EqClass (u,C) by A3, Th76; A14: rng (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) = {((((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . A),((((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . B),((((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . C),((((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . D),((((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . E),((((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . F),((((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . J),((((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . M),((((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . N)} by Th78; rng (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) c= bool Y proof let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in rng (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) or t in bool Y ) assume t in rng (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) ; ::_thesis: t in bool Y then ( t = (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . A or t = (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . B or t = (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . C or t = (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . D or t = (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . E or t = (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . F or t = (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . J or t = (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . M or t = (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . N ) by A14, ENUMSET1:def_7; hence t in bool Y by A4, A6, A13, A12, A8, A7, A10, A9, A11; ::_thesis: verum end; then reconsider FF = rng (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) as Subset-Family of Y ; A15: dom (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) = G by A2, Th77; then A in dom (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by A2, ENUMSET1:def_7; then A16: (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . A in rng (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then A17: Intersect FF = meet (rng (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A))))) by SETFAM_1:def_9; for d being set st d in G holds (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . d in d proof let d be set ; ::_thesis: ( d in G implies (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . d in d ) assume d in G ; ::_thesis: (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . d in d then ( d = A or d = B or d = C or d = D or d = E or d = F or d = J or d = M or d = N ) by A2, ENUMSET1:def_7; hence (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . d in d by A4, A6, A13, A12, A8, A7, A10, A9, A11; ::_thesis: verum end; then Intersect FF <> {} by A1, A15, BVFUNC_2:def_5; then consider m being set such that A18: m in Intersect FF by XBOOLE_0:def_1; C in dom (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by A2, A15, ENUMSET1:def_7; then (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . C in rng (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then A19: m in EqClass (u,C) by A13, A17, A18, SETFAM_1:def_1; B in dom (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by A2, A15, ENUMSET1:def_7; then (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . B in rng (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then m in EqClass (u,B) by A6, A17, A18, SETFAM_1:def_1; then A20: m in (EqClass (u,B)) /\ (EqClass (u,C)) by A19, XBOOLE_0:def_4; D in dom (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by A2, A15, ENUMSET1:def_7; then (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . D in rng (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then m in EqClass (u,D) by A12, A17, A18, SETFAM_1:def_1; then A21: m in ((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D)) by A20, XBOOLE_0:def_4; E in dom (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by A2, A15, ENUMSET1:def_7; then (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . E in rng (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then m in EqClass (u,E) by A8, A17, A18, SETFAM_1:def_1; then A22: m in (((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E)) by A21, XBOOLE_0:def_4; F in dom (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by A2, A15, ENUMSET1:def_7; then (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . F in rng (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then m in EqClass (u,F) by A7, A17, A18, SETFAM_1:def_1; then A23: m in ((((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E))) /\ (EqClass (u,F)) by A22, XBOOLE_0:def_4; J in dom (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by A2, A15, ENUMSET1:def_7; then (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . J in rng (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then m in EqClass (u,J) by A10, A17, A18, SETFAM_1:def_1; then A24: m in (((((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E))) /\ (EqClass (u,F))) /\ (EqClass (u,J)) by A23, XBOOLE_0:def_4; M in dom (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by A2, A15, ENUMSET1:def_7; then (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . M in rng (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then m in EqClass (u,M) by A9, A17, A18, SETFAM_1:def_1; then A25: m in ((((((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E))) /\ (EqClass (u,F))) /\ (EqClass (u,J))) /\ (EqClass (u,M)) by A24, XBOOLE_0:def_4; N in dom (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by A2, A15, ENUMSET1:def_7; then (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . N in rng (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then m in EqClass (u,N) by A11, A17, A18, SETFAM_1:def_1; then A26: m in (((((((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E))) /\ (EqClass (u,F))) /\ (EqClass (u,J))) /\ (EqClass (u,M))) /\ (EqClass (u,N)) by A25, XBOOLE_0:def_4; m in EqClass (z,A) by A4, A16, A17, A18, SETFAM_1:def_1; hence (EqClass (u,(((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N))) /\ (EqClass (z,A)) <> {} by A5, A26, XBOOLE_0:def_4; ::_thesis: verum end; theorem :: BVFUNC14:80 for Y being non empty set for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M, N being a_partition of Y for z, u being Element of Y st G is independent & G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N & EqClass (z,((((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)) = EqClass (u,((((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)) holds EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G))) proof let Y be non empty set ; ::_thesis: for G being Subset of (PARTITIONS Y) for A, B, C, D, E, F, J, M, N being a_partition of Y for z, u being Element of Y st G is independent & G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N & EqClass (z,((((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)) = EqClass (u,((((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)) holds EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G))) let G be Subset of (PARTITIONS Y); ::_thesis: for A, B, C, D, E, F, J, M, N being a_partition of Y for z, u being Element of Y st G is independent & G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N & EqClass (z,((((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)) = EqClass (u,((((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)) holds EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G))) let A, B, C, D, E, F, J, M, N be a_partition of Y; ::_thesis: for z, u being Element of Y st G is independent & G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N & EqClass (z,((((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)) = EqClass (u,((((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)) holds EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G))) let z, u be Element of Y; ::_thesis: ( G is independent & G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N & EqClass (z,((((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)) = EqClass (u,((((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)) implies EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G))) ) assume that A1: G is independent and A2: G = {A,B,C,D,E,F,J,M,N} and A3: ( A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N ) and A4: EqClass (z,((((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)) = EqClass (u,((((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)) ; ::_thesis: EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G))) set h = ((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A))); A5: (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . A = EqClass (z,A) by A3, Th76; set L = EqClass (z,((((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)); set GG = EqClass (u,(((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)); reconsider I = EqClass (z,A) as set ; EqClass (u,(((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)) = (EqClass (u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M))) /\ (EqClass (u,N)) by Th1; then EqClass (u,(((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)) = ((EqClass (u,(((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J))) /\ (EqClass (u,M))) /\ (EqClass (u,N)) by Th1; then EqClass (u,(((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)) = (((EqClass (u,((((B '/\' C) '/\' D) '/\' E) '/\' F))) /\ (EqClass (u,J))) /\ (EqClass (u,M))) /\ (EqClass (u,N)) by Th1; then EqClass (u,(((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)) = ((((EqClass (u,(((B '/\' C) '/\' D) '/\' E))) /\ (EqClass (u,F))) /\ (EqClass (u,J))) /\ (EqClass (u,M))) /\ (EqClass (u,N)) by Th1; then EqClass (u,(((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)) = (((((EqClass (u,((B '/\' C) '/\' D))) /\ (EqClass (u,E))) /\ (EqClass (u,F))) /\ (EqClass (u,J))) /\ (EqClass (u,M))) /\ (EqClass (u,N)) by Th1; then EqClass (u,(((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)) = ((((((EqClass (u,(B '/\' C))) /\ (EqClass (u,D))) /\ (EqClass (u,E))) /\ (EqClass (u,F))) /\ (EqClass (u,J))) /\ (EqClass (u,M))) /\ (EqClass (u,N)) by Th1; then A6: (EqClass (u,(((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N))) /\ I = ((((((((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E))) /\ (EqClass (u,F))) /\ (EqClass (u,J))) /\ (EqClass (u,M))) /\ (EqClass (u,N))) /\ (EqClass (z,A)) by Th1; A7: CompF (A,G) = ((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N by A2, A3, Th67; reconsider HH = EqClass (z,(CompF (B,G))) as set ; A8: z in HH by EQREL_1:def_6; A9: A '/\' ((((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N) = (A '/\' (((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M)) '/\' N by PARTIT1:14 .= ((A '/\' ((((C '/\' D) '/\' E) '/\' F) '/\' J)) '/\' M) '/\' N by PARTIT1:14 .= (((A '/\' (((C '/\' D) '/\' E) '/\' F)) '/\' J) '/\' M) '/\' N by PARTIT1:14 .= ((((A '/\' ((C '/\' D) '/\' E)) '/\' F) '/\' J) '/\' M) '/\' N by PARTIT1:14 .= (((((A '/\' (C '/\' D)) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N by PARTIT1:14 .= ((((((A '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N by PARTIT1:14 ; A10: (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . B = EqClass (u,B) by A3, Th76; A11: (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . N = EqClass (u,N) by A3, Th76; A12: (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . D = EqClass (u,D) by A3, Th76; A13: (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . C = EqClass (u,C) by A3, Th76; A14: (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . M = EqClass (u,M) by A3, Th76; A15: (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . J = EqClass (u,J) by A3, Th76; A16: (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . F = EqClass (u,F) by A3, Th76; A17: (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . E = EqClass (u,E) by A3, Th76; A18: rng (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) = {((((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . A),((((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . B),((((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . C),((((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . D),((((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . E),((((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . F),((((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . J),((((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . M),((((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . N)} by Th78; rng (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) c= bool Y proof let t be set ; :: according to TARSKI:def_3 ::_thesis: ( not t in rng (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) or t in bool Y ) assume t in rng (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) ; ::_thesis: t in bool Y then ( t = (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . A or t = (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . B or t = (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . C or t = (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . D or t = (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . E or t = (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . F or t = (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . J or t = (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . M or t = (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . N ) by A18, ENUMSET1:def_7; hence t in bool Y by A5, A10, A13, A12, A17, A16, A15, A14, A11; ::_thesis: verum end; then reconsider FF = rng (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) as Subset-Family of Y ; A19: dom (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) = G by A2, Th77; then A in dom (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by A2, ENUMSET1:def_7; then A20: (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . A in rng (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then A21: Intersect FF = meet (rng (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A))))) by SETFAM_1:def_9; for d being set st d in G holds (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . d in d proof let d be set ; ::_thesis: ( d in G implies (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . d in d ) assume d in G ; ::_thesis: (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . d in d then ( d = A or d = B or d = C or d = D or d = E or d = F or d = J or d = M or d = N ) by A2, ENUMSET1:def_7; hence (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . d in d by A5, A10, A13, A12, A17, A16, A15, A14, A11; ::_thesis: verum end; then Intersect FF <> {} by A1, A19, BVFUNC_2:def_5; then consider m being set such that A22: m in Intersect FF by XBOOLE_0:def_1; C in dom (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by A2, A19, ENUMSET1:def_7; then (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . C in rng (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then A23: m in EqClass (u,C) by A13, A21, A22, SETFAM_1:def_1; B in dom (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by A2, A19, ENUMSET1:def_7; then (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . B in rng (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then m in EqClass (u,B) by A10, A21, A22, SETFAM_1:def_1; then A24: m in (EqClass (u,B)) /\ (EqClass (u,C)) by A23, XBOOLE_0:def_4; D in dom (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by A2, A19, ENUMSET1:def_7; then (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . D in rng (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then m in EqClass (u,D) by A12, A21, A22, SETFAM_1:def_1; then A25: m in ((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D)) by A24, XBOOLE_0:def_4; E in dom (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by A2, A19, ENUMSET1:def_7; then (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . E in rng (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then m in EqClass (u,E) by A17, A21, A22, SETFAM_1:def_1; then A26: m in (((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E)) by A25, XBOOLE_0:def_4; F in dom (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by A2, A19, ENUMSET1:def_7; then (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . F in rng (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then m in EqClass (u,F) by A16, A21, A22, SETFAM_1:def_1; then A27: m in ((((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E))) /\ (EqClass (u,F)) by A26, XBOOLE_0:def_4; J in dom (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by A2, A19, ENUMSET1:def_7; then (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . J in rng (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then m in EqClass (u,J) by A15, A21, A22, SETFAM_1:def_1; then A28: m in (((((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E))) /\ (EqClass (u,F))) /\ (EqClass (u,J)) by A27, XBOOLE_0:def_4; M in dom (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by A2, A19, ENUMSET1:def_7; then (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . M in rng (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then m in EqClass (u,M) by A14, A21, A22, SETFAM_1:def_1; then A29: m in ((((((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E))) /\ (EqClass (u,F))) /\ (EqClass (u,J))) /\ (EqClass (u,M)) by A28, XBOOLE_0:def_4; N in dom (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by A2, A19, ENUMSET1:def_7; then (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . N in rng (((((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def_3; then m in EqClass (u,N) by A11, A21, A22, SETFAM_1:def_1; then A30: m in (((((((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E))) /\ (EqClass (u,F))) /\ (EqClass (u,J))) /\ (EqClass (u,M))) /\ (EqClass (u,N)) by A29, XBOOLE_0:def_4; m in EqClass (z,A) by A5, A20, A21, A22, SETFAM_1:def_1; then (EqClass (u,(((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N))) /\ I <> {} by A6, A30, XBOOLE_0:def_4; then consider p being set such that A31: p in (EqClass (u,(((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N))) /\ I by XBOOLE_0:def_1; reconsider p = p as Element of Y by A31; set K = EqClass (p,((((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)); A32: p in EqClass (u,(((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)) by A31, XBOOLE_0:def_4; A33: p in EqClass (p,((((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)) by EQREL_1:def_6; EqClass (u,(((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)) = EqClass (u,((((((B '/\' (C '/\' D)) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)) by PARTIT1:14; then EqClass (u,(((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)) = EqClass (u,(((((B '/\' ((C '/\' D) '/\' E)) '/\' F) '/\' J) '/\' M) '/\' N)) by PARTIT1:14; then EqClass (u,(((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)) = EqClass (u,((((B '/\' (((C '/\' D) '/\' E) '/\' F)) '/\' J) '/\' M) '/\' N)) by PARTIT1:14; then EqClass (u,(((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)) = EqClass (u,(((B '/\' ((((C '/\' D) '/\' E) '/\' F) '/\' J)) '/\' M) '/\' N)) by PARTIT1:14; then EqClass (u,(((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)) = EqClass (u,((B '/\' (((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M)) '/\' N)) by PARTIT1:14; then EqClass (u,(((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)) = EqClass (u,(B '/\' ((((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N))) by PARTIT1:14; then EqClass (u,(((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)) c= EqClass (z,((((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)) by A4, BVFUNC11:3; then EqClass (p,((((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)) meets EqClass (z,((((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)) by A32, A33, XBOOLE_0:3; then EqClass (p,((((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)) = EqClass (z,((((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)) by EQREL_1:41; then A34: z in EqClass (p,((((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)) by EQREL_1:def_6; ( p in EqClass (p,((((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)) & p in I ) by A31, EQREL_1:def_6, XBOOLE_0:def_4; then A35: p in I /\ (EqClass (p,((((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N))) by XBOOLE_0:def_4; then ( I /\ (EqClass (p,((((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N))) in INTERSECTION (A,((((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)) & not I /\ (EqClass (p,((((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N))) in {{}} ) by SETFAM_1:def_5, TARSKI:def_1; then A36: I /\ (EqClass (p,((((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N))) in (INTERSECTION (A,((((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N))) \ {{}} by XBOOLE_0:def_5; z in I by EQREL_1:def_6; then z in I /\ (EqClass (p,((((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N))) by A34, XBOOLE_0:def_4; then A37: I /\ (EqClass (p,((((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N))) meets HH by A8, XBOOLE_0:3; CompF (B,G) = ((((((A '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N by A2, A3, Th68; then I /\ (EqClass (p,((((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N))) in CompF (B,G) by A36, A9, PARTIT1:def_4; then p in HH by A35, A37, EQREL_1:def_4; hence EqClass (u,(CompF (A,G))) meets EqClass (z,(CompF (B,G))) by A7, A32, XBOOLE_0:3; ::_thesis: verum end;