:: 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;