:: Algebra of Normal Forms
:: by Andrzej Trybulec
::
:: Received October 5, 1990
:: Copyright (c) 1990-2021 Association of Mizar Users

definition

let A, B be non empty preBoolean set ;
let x, y be Element of [:A,B:];

pred x c= y means :: NORMFORM:def 1
( x `1 c= y `1 & x `2 c= y `2 );

reflexivity
for x being Element of [:A,B:] holds
( x `1 c= x `1 & x `2 c= x `2 ) ;

func x \/ y -> Element of [:A,B:] equals :: NORMFORM:def 2
[((x `1) \/ (y `1)),((x `2) \/ (y `2))];

correctness
coherence
[((x `1) \/ (y `1)),((x `2) \/ (y `2))] is Element of [:A,B:];
;
commutativity
for b₁, x, y being Element of [:A,B:] st b₁ = [((x `1) \/ (y `1)),((x `2) \/ (y `2))] holds
b₁ = [((y `1) \/ (x `1)),((y `2) \/ (x `2))] ;
idempotence
for x being Element of [:A,B:] holds x = [((x `1) \/ (x `1)),((x `2) \/ (x `2))] by MCART_1:21;

func x /\ y -> Element of [:A,B:] equals :: NORMFORM:def 3
[((x `1) /\ (y `1)),((x `2) /\ (y `2))];

correctness
coherence
[((x `1) /\ (y `1)),((x `2) /\ (y `2))] is Element of [:A,B:];
;
commutativity
for b₁, x, y being Element of [:A,B:] st b₁ = [((x `1) /\ (y `1)),((x `2) /\ (y `2))] holds
b₁ = [((y `1) /\ (x `1)),((y `2) /\ (x `2))] ;
idempotence
for x being Element of [:A,B:] holds x = [((x `1) /\ (x `1)),((x `2) /\ (x `2))] by MCART_1:21;

func x \ y -> Element of [:A,B:] equals :: NORMFORM:def 4
[((x `1) \ (y `1)),((x `2) \ (y `2))];

correctness
coherence
[((x `1) \ (y `1)),((x `2) \ (y `2))] is Element of [:A,B:];
;

func x \+\ y -> Element of [:A,B:] equals :: NORMFORM:def 5
[((x `1) \+\ (y `1)),((x `2) \+\ (y `2))];

correctness
coherence
[((x `1) \+\ (y `1)),((x `2) \+\ (y `2))] is Element of [:A,B:];
;
commutativity
for b₁, x, y being Element of [:A,B:] st b₁ = [((x `1) \+\ (y `1)),((x `2) \+\ (y `2))] holds
b₁ = [((y `1) \+\ (x `1)),((y `2) \+\ (x `2))] ;

end;

:: deftheorem defines c= NORMFORM:def 1 :
for A, B being non empty preBoolean set
for x, y being Element of [:A,B:] holds
( x c= y iff ( x `1 c= y `1 & x `2 c= y `2 ) );

:: deftheorem defines \/ NORMFORM:def 2 :
for A, B being non empty preBoolean set
for x, y being Element of [:A,B:] holds x \/ y = [((x `1) \/ (y `1)),((x `2) \/ (y `2))];

:: deftheorem defines /\ NORMFORM:def 3 :
for A, B being non empty preBoolean set
for x, y being Element of [:A,B:] holds x /\ y = [((x `1) /\ (y `1)),((x `2) /\ (y `2))];

:: deftheorem defines \ NORMFORM:def 4 :
for A, B being non empty preBoolean set
for x, y being Element of [:A,B:] holds x \ y = [((x `1) \ (y `1)),((x `2) \ (y `2))];

:: deftheorem defines \+\ NORMFORM:def 5 :
for A, B being non empty preBoolean set
for x, y being Element of [:A,B:] holds x \+\ y = [((x `1) \+\ (y `1)),((x `2) \+\ (y `2))];

theorem Th1: :: NORMFORM:1

for A, B being non empty preBoolean set
for a, b being Element of [:A,B:] st a c= b & b c= a holds
a = b

proof end;

theorem Th2: :: NORMFORM:2

for A, B being non empty preBoolean set
for a, b, c being Element of [:A,B:] st a c= b & b c= c holds
a c= c

proof end;

theorem Th3: :: NORMFORM:3

for A, B being non empty preBoolean set
for a, b, c being Element of [:A,B:] holds (a \/ b) \/ c = a \/ (b \/ c)

proof end;

theorem :: NORMFORM:4

for A, B being non empty preBoolean set
for a, b, c being Element of [:A,B:] holds (a /\ b) /\ c = a /\ (b /\ c)

proof end;

theorem Th5: :: NORMFORM:5

for A, B being non empty preBoolean set
for a, b, c being Element of [:A,B:] holds a /\ (b \/ c) = (a /\ b) \/ (a /\ c)

proof end;

theorem Th6: :: NORMFORM:6

for A, B being non empty preBoolean set
for a, b being Element of [:A,B:] holds a \/ (b /\ a) = a

proof end;

theorem Th7: :: NORMFORM:7

for A, B being non empty preBoolean set
for a, b being Element of [:A,B:] holds a /\ (b \/ a) = a

proof end;

theorem :: NORMFORM:8

for A, B being non empty preBoolean set
for a, b, c being Element of [:A,B:] holds a \/ (b /\ c) = (a \/ b) /\ (a \/ c)

proof end;

theorem Th9: :: NORMFORM:9

for A, B being non empty preBoolean set
for a, b, c being Element of [:A,B:] st a c= c & b c= c holds
a \/ b c= c by XBOOLE_1:8;

theorem Th10: :: NORMFORM:10

for A, B being non empty preBoolean set
for a, b being Element of [:A,B:] holds
( a c= a \/ b & b c= a \/ b ) by XBOOLE_1:7;

theorem :: NORMFORM:11

for A, B being non empty preBoolean set
for a, b being Element of [:A,B:] st a = a \/ b holds
b c= a by Th10;

theorem Th12: :: NORMFORM:12

for A, B being non empty preBoolean set
for a, b, c being Element of [:A,B:] st a c= b holds
( c \/ a c= c \/ b & a \/ c c= b \/ c ) by XBOOLE_1:9;

theorem :: NORMFORM:13

for A, B being non empty preBoolean set
for a, b being Element of [:A,B:] holds (a \ b) \/ b = a \/ b

proof end;

theorem :: NORMFORM:14

for A, B being non empty preBoolean set
for a, b, c being Element of [:A,B:] st a \ b c= c holds
a c= b \/ c by XBOOLE_1:44;

theorem :: NORMFORM:15

for A, B being non empty preBoolean set
for a, b, c being Element of [:A,B:] st a c= b \/ c holds
a \ c c= b by XBOOLE_1:43;

definition

let A be set ;

func FinPairUnion A -> BinOp of [:(Fin A),(Fin A):] means :Def6: :: NORMFORM:def 6
for x, y being Element of [:(Fin A),(Fin A):] holds it . (x,y) = x \/ y;

existence
ex b₁ being BinOp of [:(Fin A),(Fin A):] st
for x, y being Element of [:(Fin A),(Fin A):] holds b₁ . (x,y) = x \/ y

proof end;

uniqueness
for b₁, b₂ being BinOp of [:(Fin A),(Fin A):] st ( for x, y being Element of [:(Fin A),(Fin A):] holds b₁ . (x,y) = x \/ y ) & ( for x, y being Element of [:(Fin A),(Fin A):] holds b₂ . (x,y) = x \/ y ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def6 defines FinPairUnion NORMFORM:def 6 :
for A being set
for b₂ being BinOp of [:(Fin A),(Fin A):] holds
( b₂ = FinPairUnion A iff for x, y being Element of [:(Fin A),(Fin A):] holds b₂ . (x,y) = x \/ y );

definition

let X be non empty set ;
let A be set ;
let B be Element of Fin X;
let f be Function of X,[:(Fin A),(Fin A):];

func FinPairUnion (B,f) -> Element of [:(Fin A),(Fin A):] equals :: NORMFORM:def 7
(FinPairUnion A) $$ (B,f);

correctness
coherence
(FinPairUnion A) $$ (B,f) is Element of [:(Fin A),(Fin A):];
;

end;

:: deftheorem defines FinPairUnion NORMFORM:def 7 :
for X being non empty set
for A being set
for B being Element of Fin X
for f being Function of X,[:(Fin A),(Fin A):] holds FinPairUnion (B,f) = (FinPairUnion A) $$ (B,f);

Lm1: for A being set holds FinPairUnion A is idempotent

proof end;

Lm2: for A being set holds FinPairUnion A is commutative

proof end;

Lm3: for A being set holds FinPairUnion A is associative

proof end;

registration

let A be set ;

cluster FinPairUnion A -> commutative associative idempotent ;

coherence
( FinPairUnion A is commutative & FinPairUnion A is idempotent & FinPairUnion A is associative ) by Lm1, Lm2, Lm3;

end;

theorem :: NORMFORM:16

for A being set
for X being non empty set
for f being Function of X,[:(Fin A),(Fin A):]
for B being Element of Fin X
for x being Element of X st x in B holds
f . x c= FinPairUnion (B,f)

proof end;

theorem Th17: :: NORMFORM:17

for A being set holds [({}. A),({}. A)] is_a_unity_wrt FinPairUnion A

proof end;

theorem Th18: :: NORMFORM:18

for A being set holds FinPairUnion A is having_a_unity

proof end;

theorem Th19: :: NORMFORM:19

for A being set holds the_unity_wrt (FinPairUnion A) = [({}. A),({}. A)]

proof end;

theorem Th20: :: NORMFORM:20

for A being set
for x being Element of [:(Fin A),(Fin A):] holds the_unity_wrt (FinPairUnion A) c= x

proof end;

theorem :: NORMFORM:21

for A being set
for X being non empty set
for f being Function of X,[:(Fin A),(Fin A):]
for B being Element of Fin X
for c being Element of [:(Fin A),(Fin A):] st ( for x being Element of X st x in B holds
f . x c= c ) holds
FinPairUnion (B,f) c= c

proof end;

theorem :: NORMFORM:22

for A being set
for X being non empty set
for B being Element of Fin X
for f, g being Function of X,[:(Fin A),(Fin A):] st f | B = g | B holds
FinPairUnion (B,f) = FinPairUnion (B,g)

proof end;

:: Part 2. Disjoint pairs of finite sets

definition

let X be set ;

func DISJOINT_PAIRS X -> Subset of [:(Fin X),(Fin X):] equals :: NORMFORM:def 8
{ a where a is Element of [:(Fin X),(Fin X):] : a `1 misses a `2 } ;

coherence
{ a where a is Element of [:(Fin X),(Fin X):] : a `1 misses a `2 } is Subset of [:(Fin X),(Fin X):]

proof end;

end;

:: deftheorem defines DISJOINT_PAIRS NORMFORM:def 8 :
for X being set holds DISJOINT_PAIRS X = { a where a is Element of [:(Fin X),(Fin X):] : a `1 misses a `2 } ;

registration

let X be set ;

cluster DISJOINT_PAIRS X -> non empty ;

coherence
not DISJOINT_PAIRS X is empty

proof end;

end;

theorem Th23: :: NORMFORM:23

for X being set
for y being Element of [:(Fin X),(Fin X):] holds
( y in DISJOINT_PAIRS X iff y `1 misses y `2 )

proof end;

theorem :: NORMFORM:24

for X being set
for x, y being Element of [:(Fin X),(Fin X):] st y in DISJOINT_PAIRS X & x in DISJOINT_PAIRS X holds
( y \/ x in DISJOINT_PAIRS X iff ((y `1) /\ (x `2)) \/ ((x `1) /\ (y `2)) = {} )

proof end;

theorem :: NORMFORM:25

for X being set
for a being Element of DISJOINT_PAIRS X holds a `1 misses a `2 by Th23;

theorem Th26: :: NORMFORM:26

for X being set
for x being Element of [:(Fin X),(Fin X):]
for b being Element of DISJOINT_PAIRS X st x c= b holds
x is Element of DISJOINT_PAIRS X

proof end;

theorem Th27: :: NORMFORM:27

for X being set
for a being Element of DISJOINT_PAIRS X
for x being set holds
( not x in a `1 or not x in a `2 ) by Th23, XBOOLE_0:3;

theorem :: NORMFORM:28

for X being set
for a, b being Element of DISJOINT_PAIRS X st not a \/ b in DISJOINT_PAIRS X holds
ex p being Element of X st
( ( p in a `1 & p in b `2 ) or ( p in b `1 & p in a `2 ) )

proof end;

theorem :: NORMFORM:29

for X being set
for x being Element of [:(Fin X),(Fin X):] st x `1 misses x `2 holds
x is Element of DISJOINT_PAIRS X by Th23;

theorem :: NORMFORM:30

for X being set
for a being Element of DISJOINT_PAIRS X
for V, W being set st V c= a `1 & W c= a `2 holds
[V,W] is Element of DISJOINT_PAIRS X

proof end;

Lm4: for A being set holds {} in { B where B is Element of Fin (DISJOINT_PAIRS A) : for a, b being Element of DISJOINT_PAIRS A st a in B & b in B & a c= b holds
a = b }

proof end;

definition

let A be set ;

func Normal_forms_on A -> Subset of (Fin (DISJOINT_PAIRS A)) equals :: NORMFORM:def 9
{ B where B is Element of Fin (DISJOINT_PAIRS A) : for a, b being Element of DISJOINT_PAIRS A st a in B & b in B & a c= b holds
a = b } ;

coherence
{ B where B is Element of Fin (DISJOINT_PAIRS A) : for a, b being Element of DISJOINT_PAIRS A st a in B & b in B & a c= b holds
a = b } is Subset of (Fin (DISJOINT_PAIRS A))

proof end;

end;

:: deftheorem defines Normal_forms_on NORMFORM:def 9 :
for A being set holds Normal_forms_on A = { B where B is Element of Fin (DISJOINT_PAIRS A) : for a, b being Element of DISJOINT_PAIRS A st a in B & b in B & a c= b holds
a = b } ;

registration

let A be set ;

cluster Normal_forms_on A -> non empty ;

coherence
not Normal_forms_on A is empty by Lm4;

end;

theorem :: NORMFORM:31

for A being set holds {} in Normal_forms_on A by Lm4;

theorem Th32: :: NORMFORM:32

for A being set
for a, b being Element of DISJOINT_PAIRS A
for B being Element of Fin (DISJOINT_PAIRS A) st B in Normal_forms_on A & a in B & b in B & a c= b holds
a = b

proof end;

theorem Th33: :: NORMFORM:33

for A being set
for B being Element of Fin (DISJOINT_PAIRS A) st ( for a, b being Element of DISJOINT_PAIRS A st a in B & b in B & a c= b holds
a = b ) holds
B in Normal_forms_on A ;

definition

let A be set ;
let B be Element of Fin (DISJOINT_PAIRS A);

func mi B -> Element of Normal_forms_on A equals :: NORMFORM:def 10
{ t where t is Element of DISJOINT_PAIRS A : for s being Element of DISJOINT_PAIRS A holds
( ( s in B & s c= t ) iff s = t ) } ;

coherence
{ t where t is Element of DISJOINT_PAIRS A : for s being Element of DISJOINT_PAIRS A holds
( ( s in B & s c= t ) iff s = t ) } is Element of Normal_forms_on A

proof end;

correctness
;
let C be Element of Fin (DISJOINT_PAIRS A);

func B ^ C -> Element of Fin (DISJOINT_PAIRS A) equals :: NORMFORM:def 11
(DISJOINT_PAIRS A) /\ { (s \/ t) where s, t is Element of DISJOINT_PAIRS A : ( s in B & t in C ) } ;

coherence
(DISJOINT_PAIRS A) /\ { (s \/ t) where s, t is Element of DISJOINT_PAIRS A : ( s in B & t in C ) } is Element of Fin (DISJOINT_PAIRS A)

proof end;

correctness
;

end;

:: deftheorem defines mi NORMFORM:def 10 :
for A being set
for B being Element of Fin (DISJOINT_PAIRS A) holds mi B = { t where t is Element of DISJOINT_PAIRS A : for s being Element of DISJOINT_PAIRS A holds
( ( s in B & s c= t ) iff s = t ) } ;

:: deftheorem defines ^ NORMFORM:def 11 :
for A being set
for B, C being Element of Fin (DISJOINT_PAIRS A) holds B ^ C = (DISJOINT_PAIRS A) /\ { (s \/ t) where s, t is Element of DISJOINT_PAIRS A : ( s in B & t in C ) } ;

theorem Th34: :: NORMFORM:34

for A being set
for x being Element of [:(Fin A),(Fin A):]
for B, C being Element of Fin (DISJOINT_PAIRS A) st x in B ^ C holds
ex b, c being Element of DISJOINT_PAIRS A st
( b in B & c in C & x = b \/ c )

proof end;

theorem Th35: :: NORMFORM:35

for A being set
for b, c being Element of DISJOINT_PAIRS A
for B, C being Element of Fin (DISJOINT_PAIRS A) st b in B & c in C & b \/ c in DISJOINT_PAIRS A holds
b \/ c in B ^ C

proof end;

theorem Th36: :: NORMFORM:36

for A being set
for a, b being Element of DISJOINT_PAIRS A
for B being Element of Fin (DISJOINT_PAIRS A) st a in mi B holds
( a in B & ( b in B & b c= a implies b = a ) )

proof end;

theorem :: NORMFORM:37

for A being set
for a being Element of DISJOINT_PAIRS A
for B being Element of Fin (DISJOINT_PAIRS A) st a in mi B holds
a in B by Th36;

theorem :: NORMFORM:38

for A being set
for a, b being Element of DISJOINT_PAIRS A
for B being Element of Fin (DISJOINT_PAIRS A) st a in mi B & b in B & b c= a holds
b = a by Th36;

theorem Th39: :: NORMFORM:39

for A being set
for a being Element of DISJOINT_PAIRS A
for B being Element of Fin (DISJOINT_PAIRS A) st a in B & ( for b being Element of DISJOINT_PAIRS A st b in B & b c= a holds
b = a ) holds
a in mi B

proof end;

Lm5: for A being set
for B, C being Element of Fin (DISJOINT_PAIRS A) st ( for a being Element of DISJOINT_PAIRS A st a in B holds
a in C ) holds
B c= C

proof end;

theorem Th40: :: NORMFORM:40

for A being set
for B being Element of Fin (DISJOINT_PAIRS A) holds mi B c= B

proof end;

theorem Th41: :: NORMFORM:41

for A being set
for b being Element of DISJOINT_PAIRS A
for B being Element of Fin (DISJOINT_PAIRS A) st b in B holds
ex c being Element of DISJOINT_PAIRS A st
( c c= b & c in mi B )

proof end;

theorem Th42: :: NORMFORM:42

for A being set
for K being Element of Normal_forms_on A holds mi K = K

proof end;

theorem Th43: :: NORMFORM:43

for A being set
for B, C being Element of Fin (DISJOINT_PAIRS A) holds mi (B \/ C) c= (mi B) \/ C

proof end;

theorem Th44: :: NORMFORM:44

for A being set
for B, C being Element of Fin (DISJOINT_PAIRS A) holds mi ((mi B) \/ C) = mi (B \/ C)

proof end;

theorem :: NORMFORM:45

for A being set
for B, C being Element of Fin (DISJOINT_PAIRS A) holds mi (B \/ (mi C)) = mi (B \/ C) by Th44;

theorem Th46: :: NORMFORM:46

for A being set
for B, C, D being Element of Fin (DISJOINT_PAIRS A) st B c= C holds
B ^ D c= C ^ D

proof end;

Lm6: for A being set
for a being Element of DISJOINT_PAIRS A
for B, C being Element of Fin (DISJOINT_PAIRS A) st a in B ^ C holds
ex b being Element of DISJOINT_PAIRS A st
( b c= a & b in (mi B) ^ C )

proof end;

theorem Th47: :: NORMFORM:47

for A being set
for B, C being Element of Fin (DISJOINT_PAIRS A) holds mi (B ^ C) c= (mi B) ^ C

proof end;

theorem Th48: :: NORMFORM:48

for A being set
for B, C being Element of Fin (DISJOINT_PAIRS A) holds B ^ C = C ^ B

proof end;

theorem Th49: :: NORMFORM:49

for A being set
for B, C, D being Element of Fin (DISJOINT_PAIRS A) st B c= C holds
D ^ B c= D ^ C

proof end;

theorem Th50: :: NORMFORM:50

for A being set
for B, C being Element of Fin (DISJOINT_PAIRS A) holds mi ((mi B) ^ C) = mi (B ^ C)

proof end;

theorem Th51: :: NORMFORM:51

for A being set
for B, C being Element of Fin (DISJOINT_PAIRS A) holds mi (B ^ (mi C)) = mi (B ^ C)

proof end;

theorem Th52: :: NORMFORM:52

for A being set
for K, L, M being Element of Normal_forms_on A holds K ^ (L ^ M) = (K ^ L) ^ M

proof end;

theorem Th53: :: NORMFORM:53

for A being set
for K, L, M being Element of Normal_forms_on A holds K ^ (L \/ M) = (K ^ L) \/ (K ^ M)

proof end;

Lm7: for A being set
for a being Element of DISJOINT_PAIRS A
for B, C being Element of Fin (DISJOINT_PAIRS A) st a in B ^ C holds
ex c being Element of DISJOINT_PAIRS A st
( c in C & c c= a )

proof end;

Lm8: for A being set
for K, L being Element of Normal_forms_on A holds mi ((K ^ L) \/ L) = mi L

proof end;

theorem Th54: :: NORMFORM:54

for A being set
for B being Element of Fin (DISJOINT_PAIRS A) holds B c= B ^ B

proof end;

theorem Th55: :: NORMFORM:55

for A being set
for K being Element of Normal_forms_on A holds mi (K ^ K) = mi K

proof end;

definition

let A be set ;

func NormForm A -> strict LattStr means :Def12: :: NORMFORM:def 12
( the carrier of it = Normal_forms_on A & ( for B, C being Element of Normal_forms_on A holds
( the L_join of it . (B,C) = mi (B \/ C) & the L_meet of it . (B,C) = mi (B ^ C) ) ) );

existence
ex b₁ being strict LattStr st
( the carrier of b₁ = Normal_forms_on A & ( for B, C being Element of Normal_forms_on A holds
( the L_join of b₁ . (B,C) = mi (B \/ C) & the L_meet of b₁ . (B,C) = mi (B ^ C) ) ) )

proof end;

uniqueness
for b₁, b₂ being strict LattStr st the carrier of b₁ = Normal_forms_on A & ( for B, C being Element of Normal_forms_on A holds
( the L_join of b₁ . (B,C) = mi (B \/ C) & the L_meet of b₁ . (B,C) = mi (B ^ C) ) ) & the carrier of b₂ = Normal_forms_on A & ( for B, C being Element of Normal_forms_on A holds
( the L_join of b₂ . (B,C) = mi (B \/ C) & the L_meet of b₂ . (B,C) = mi (B ^ C) ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def12 defines NormForm NORMFORM:def 12 :
for A being set
for b₂ being strict LattStr holds
( b₂ = NormForm A iff ( the carrier of b₂ = Normal_forms_on A & ( for B, C being Element of Normal_forms_on A holds
( the L_join of b₂ . (B,C) = mi (B \/ C) & the L_meet of b₂ . (B,C) = mi (B ^ C) ) ) ) );

registration

let A be set ;

cluster NormForm A -> non empty strict ;

coherence
not NormForm A is empty

proof end;

end;

Lm9: for A being set
for a, b being Element of (NormForm A) holds a "\/" b = b "\/" a

proof end;

Lm10: for A being set
for a, b, c being Element of (NormForm A) holds a "\/" (b "\/" c) = (a "\/" b) "\/" c

proof end;

Lm11: for A being set
for K, L being Element of Normal_forms_on A holds the L_join of (NormForm A) . (( the L_meet of (NormForm A) . (K,L)),L) = L

proof end;

Lm12: for A being set
for a, b being Element of (NormForm A) holds (a "/\" b) "\/" b = b

proof end;

Lm13: for A being set
for a, b being Element of (NormForm A) holds a "/\" b = b "/\" a

proof end;

Lm14: for A being set
for a, b, c being Element of (NormForm A) holds a "/\" (b "/\" c) = (a "/\" b) "/\" c

proof end;

Lm15: for A being set
for K, L, M being Element of Normal_forms_on A holds the L_meet of (NormForm A) . (K,( the L_join of (NormForm A) . (L,M))) = the L_join of (NormForm A) . (( the L_meet of (NormForm A) . (K,L)),( the L_meet of (NormForm A) . (K,M)))

proof end;

Lm16: for A being set
for a, b being Element of (NormForm A) holds a "/\" (a "\/" b) = a

proof end;

registration

let A be set ;

cluster NormForm A -> strict Lattice-like ;

coherence
NormForm A is Lattice-like

proof end;

end;

registration

let A be set ;

cluster NormForm A -> strict distributive lower-bounded ;

coherence
( NormForm A is distributive & NormForm A is lower-bounded )

proof end;

end;

theorem :: NORMFORM:56

for A being set holds {} is Element of (NormForm A)

proof end;

theorem :: NORMFORM:57

for A being set holds Bottom (NormForm A) = {}

proof end;