let Y be non empty set ; :: thesis: for G being Subset of ()
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 (); :: 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
per cases ( B = C or B = D or C = D or ( B <> C & B <> D & C <> D ) ) ;
suppose A5: B = C ; :: thesis: CompF (A,G) = (B '/\' C) '/\' D
then G = {B,B,A,D} by
.= {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 ;
:: thesis: verum
end;
suppose A6: B = D ; :: thesis: CompF (A,G) = (B '/\' C) '/\' D
then G = {B,B,A,C} by
.= {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
.= (B '/\' C) '/\' D by PARTIT1:14 ;
:: thesis: verum
end;
suppose A7: C = D ; :: thesis: CompF (A,G) = (B '/\' C) '/\' D
then G = {C,C,A,B} by
.= {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
.= (B '/\' C) '/\' D by PARTIT1:14 ;
:: thesis: verum
end;
suppose A8: ( B <> C & B <> D & C <> D ) ; :: thesis: CompF (A,G) = (B '/\' C) '/\' D
G \ {A} = ({A} \/ {B,C,D}) \ {A} by ;
then A9: G \ {A} = ({A} \ {A}) \/ ({B,C,D} \ {A}) by XBOOLE_1:42;
A10: not B in {A} by ;
A11: ( not C in {A} & not D in {A} ) by ;
{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
.= {B} \/ {C,D} by
.= {B,C,D} by ENUMSET1:2 ;
then A12: G \ {A} = {} \/ {B,C,D} by
.= {B,C,D} ;
A13: (B '/\' C) '/\' D c= '/\' (G \ {A})
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in (B '/\' C) '/\' D or x in '/\' (G \ {A}) )
reconsider xx = x as set by TARSKI:1;
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 ;
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 ;
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,C,D) --> (b,c,d);
A22: ((B,C,D) --> (b,c,d)) . D = d by FUNCT_7:94;
A23: ((B,C,D) --> (b,c,d)) . C = c by ;
A24: rng ((B,C,D) --> (b,c,d)) = {(((B,C,D) --> (b,c,d)) . B),(((B,C,D) --> (b,c,d)) . C),(((B,C,D) --> (b,c,d)) . D)} by Lm2
.= {(((B,C,D) --> (b,c,d)) . D),(((B,C,D) --> (b,c,d)) . B),(((B,C,D) --> (b,c,d)) . C)} by ENUMSET1:59 ;
A25: ((B,C,D) --> (b,c,d)) . B = b by ;
rng ((B,C,D) --> (b,c,d)) c= bool Y
proof
let t be object ; :: according to TARSKI:def 3 :: thesis: ( not t in rng ((B,C,D) --> (b,c,d)) or t in bool Y )
assume A26: t in rng ((B,C,D) --> (b,c,d)) ; :: thesis: t in bool Y
now :: thesis: ( ( t = ((B,C,D) --> (b,c,d)) . D & t in bool Y ) or ( t = ((B,C,D) --> (b,c,d)) . B & t in bool Y ) or ( t = ((B,C,D) --> (b,c,d)) . C & t in bool Y ) )
per cases ( t = ((B,C,D) --> (b,c,d)) . D or t = ((B,C,D) --> (b,c,d)) . B or t = ((B,C,D) --> (b,c,d)) . C ) by ;
case t = ((B,C,D) --> (b,c,d)) . D ; :: thesis: t in bool Y
hence t in bool Y by ; :: thesis: verum
end;
case t = ((B,C,D) --> (b,c,d)) . B ; :: thesis: t in bool Y
hence t in bool Y by ; :: thesis: verum
end;
case t = ((B,C,D) --> (b,c,d)) . C ; :: thesis: t in bool Y
hence t in bool Y by ; :: thesis: verum
end;
end;
end;
hence t in bool Y ; :: thesis: verum
end;
then reconsider F = rng ((B,C,D) --> (b,c,d)) as Subset-Family of Y ;
A27: xx c= Intersect F
proof
let u be object ; :: according to TARSKI:def 3 :: thesis: ( not u in xx or u in Intersect F )
assume A28: u in xx ; :: thesis:
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,C,D) --> (b,c,d)) . D & u in y ) or ( y = ((B,C,D) --> (b,c,d)) . B & u in y ) or ( y = ((B,C,D) --> (b,c,d)) . C & u in y ) )
per cases ( y = ((B,C,D) --> (b,c,d)) . D or y = ((B,C,D) --> (b,c,d)) . B or y = ((B,C,D) --> (b,c,d)) . C ) by ;
case y = ((B,C,D) --> (b,c,d)) . D ; :: thesis: u in y
hence u in y by ; :: thesis: verum
end;
case A30: y = ((B,C,D) --> (b,c,d)) . B ; :: thesis: u in y
u in b /\ (c /\ d) by ;
hence u in y by ; :: thesis: verum
end;
case A31: y = ((B,C,D) --> (b,c,d)) . C ; :: thesis: u in y
u in c /\ (b /\ d) by ;
hence u in y by ; :: thesis: verum
end;
end;
end;
hence u in y ; :: thesis: verum
end;
then u in meet F by ;
hence u in Intersect F by ; :: thesis: verum
end;
A32: for p being set st p in G \ {A} holds
((B,C,D) --> (b,c,d)) . p in p
proof
let p be set ; :: thesis: ( p in G \ {A} implies ((B,C,D) --> (b,c,d)) . p in p )
assume A33: p in G \ {A} ; :: thesis: ((B,C,D) --> (b,c,d)) . p in p
now :: thesis: ( ( p = D & ((B,C,D) --> (b,c,d)) . p in p ) or ( p = B & ((B,C,D) --> (b,c,d)) . p in p ) or ( p = C & ((B,C,D) --> (b,c,d)) . p in p ) )
per cases ( p = D or p = B or p = C ) by ;
case p = D ; :: thesis: ((B,C,D) --> (b,c,d)) . p in p
hence ((B,C,D) --> (b,c,d)) . p in p by ; :: thesis: verum
end;
case p = B ; :: thesis: ((B,C,D) --> (b,c,d)) . p in p
hence ((B,C,D) --> (b,c,d)) . p in p by ; :: thesis: verum
end;
case p = C ; :: thesis: ((B,C,D) --> (b,c,d)) . p in p
hence ((B,C,D) --> (b,c,d)) . p in p by A8, A20, Lm1; :: thesis: verum
end;
end;
end;
hence ((B,C,D) --> (b,c,d)) . p in p ; :: thesis: verum
end;
A34: dom ((B,C,D) --> (b,c,d)) = {B,C,D} by FUNCT_4:128;
then D in dom ((B,C,D) --> (b,c,d)) by ENUMSET1:def 1;
then A35: rng ((B,C,D) --> (b,c,d)) <> {} by FUNCT_1:3;
Intersect F c= xx
proof
let t be object ; :: according to TARSKI:def 3 :: thesis: ( not t in Intersect F or t in xx )
assume t in Intersect F ; :: thesis: t in xx
then A36: t in meet (rng ((B,C,D) --> (b,c,d))) by ;
((B,C,D) --> (b,c,d)) . D in rng ((B,C,D) --> (b,c,d)) by ;
then A37: t in ((B,C,D) --> (b,c,d)) . D by ;
((B,C,D) --> (b,c,d)) . C in rng ((B,C,D) --> (b,c,d)) by ;
then A38: t in ((B,C,D) --> (b,c,d)) . C by ;
((B,C,D) --> (b,c,d)) . B in rng ((B,C,D) --> (b,c,d)) by ;
then t in ((B,C,D) --> (b,c,d)) . B by ;
then t in b /\ c by ;
hence t in xx by ; :: thesis: verum
end;
then x = Intersect F by ;
hence x in '/\' (G \ {A}) by ; :: thesis: verum
end;
'/\' (G \ {A}) c= (B '/\' C) '/\' D
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in '/\' (G \ {A}) or x in (B '/\' C) '/\' D )
reconsider xx = x as set by TARSKI:1;
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 ;
then A44: h . D in rng h by FUNCT_1:def 3;
set m = (h . B) /\ (h . C);
B in dom h by ;
then A45: h . B in rng h by FUNCT_1:def 3;
C in dom h by ;
then A46: h . C in rng h by FUNCT_1:def 3;
A47: xx c= ((h . B) /\ (h . C)) /\ (h . D)
proof
let m be object ; :: according to TARSKI:def 3 :: thesis: ( not m in xx or m in ((h . B) /\ (h . C)) /\ (h . D) )
assume m in xx ; :: thesis: m in ((h . B) /\ (h . C)) /\ (h . D)
then A48: m in meet (rng h) by ;
then ( m in h . B & m in h . C ) by ;
then A49: m in (h . B) /\ (h . C) by XBOOLE_0:def 4;
m in h . D by ;
hence m in ((h . B) /\ (h . C)) /\ (h . D) by ; :: 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 ;
then A51: h . D in D by A41;
A52: not x in by ;
C in G \ {A} by ;
then A53: h . C in C by A41;
B in G \ {A} by ;
then h . B in B by A41;
then (h . B) /\ (h . C) in INTERSECTION (B,C) by ;
then (h . B) /\ (h . C) in (INTERSECTION (B,C)) \ by ;
then A54: (h . B) /\ (h . C) in B '/\' C by PARTIT1:def 4;
((h . B) /\ (h . C)) /\ (h . D) c= xx
proof
let m be object ; :: according to TARSKI:def 3 :: thesis: ( not m in ((h . B) /\ (h . C)) /\ (h . D) or m in xx )
assume A55: m in ((h . B) /\ (h . C)) /\ (h . D) ; :: thesis: m in xx
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 object ; :: 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 object 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)} ) )
per cases ( x1 = B or x1 = C or x1 = D ) by ;
case x1 = B ; :: thesis: u in {(h . B),(h . C),(h . D)}
hence u in {(h . B),(h . C),(h . D)} by ; :: 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 ; :: 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 ; :: 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 ) )
per cases ( y = h . B or y = h . C or y = h . D ) by ;
case y = h . B ; :: thesis: m in y
hence m in y by ; :: thesis: verum
end;
case y = h . C ; :: thesis: m in y
hence m in y by ; :: thesis: verum
end;
case y = h . D ; :: thesis: m in y
hence m in y by ; :: thesis: verum
end;
end;
end;
hence m in y ; :: thesis: verum
end;
then m in meet (rng h) by ;
hence m in xx by ; :: thesis: verum
end;
then ((h . B) /\ (h . C)) /\ (h . D) = x by ;
then x in INTERSECTION ((B '/\' C),D) by ;
then x in (INTERSECTION ((B '/\' C),D)) \ by ;
hence x in (B '/\' C) '/\' D by PARTIT1:def 4; :: thesis: verum
end;
then '/\' (G \ {A}) = (B '/\' C) '/\' D by ;
hence CompF (A,G) = (B '/\' C) '/\' D by BVFUNC_2:def 7; :: thesis: verum
end;
end;