:: Duality in Relation Structures
:: by Grzegorz Bancerek
::
:: Received November 12, 1996
:: Copyright (c) 1996 Association of Mizar Users
theorem Th1: :: YELLOW_7:1
theorem Th2: :: YELLOW_7:2
theorem :: YELLOW_7:3
theorem Th4: :: YELLOW_7:4
theorem Th5: :: YELLOW_7:5
theorem Th6: :: YELLOW_7:6
theorem Th7: :: YELLOW_7:7
theorem Th8: :: YELLOW_7:8
theorem Th9: :: YELLOW_7:9
theorem Th10: :: YELLOW_7:10
theorem Th11: :: YELLOW_7:11
theorem Th12: :: YELLOW_7:12
theorem Th13: :: YELLOW_7:13
theorem Th14: :: YELLOW_7:14
theorem :: YELLOW_7:15
theorem Th16: :: YELLOW_7:16
theorem Th17: :: YELLOW_7:17
theorem :: YELLOW_7:18
theorem Th19: :: YELLOW_7:19
theorem :: YELLOW_7:20
theorem Th21: :: YELLOW_7:21
theorem Th22: :: YELLOW_7:22
theorem Th23: :: YELLOW_7:23
theorem Th24: :: YELLOW_7:24
theorem Th25: :: YELLOW_7:25
theorem Th26: :: YELLOW_7:26
theorem :: YELLOW_7:27
theorem Th28: :: YELLOW_7:28
theorem :: YELLOW_7:29
theorem Th30: :: YELLOW_7:30
theorem Th31: :: YELLOW_7:31
theorem :: YELLOW_7:32
theorem :: YELLOW_7:33
theorem :: YELLOW_7:34
theorem Th35: :: YELLOW_7:35
theorem Th36: :: YELLOW_7:36
theorem :: YELLOW_7:37
:: deftheorem Def1 defines ComplMap YELLOW_7:def 1 :
theorem :: YELLOW_7:38
theorem :: YELLOW_7:39
for
S,
T being non
empty RelStr for
f being
set holds
( (
f is
Function of
S,
T implies
f is
Function of
(S opp ),
T ) & (
f is
Function of
(S opp ),
T implies
f is
Function of
S,
T ) & (
f is
Function of
S,
T implies
f is
Function of
S,
(T opp ) ) & (
f is
Function of
S,
(T opp ) implies
f is
Function of
S,
T ) & (
f is
Function of
S,
T implies
f is
Function of
(S opp ),
(T opp ) ) & (
f is
Function of
(S opp ),
(T opp ) implies
f is
Function of
S,
T ) ) ;
theorem :: YELLOW_7:40
theorem :: YELLOW_7:41
theorem Th42: :: YELLOW_7:42
theorem :: YELLOW_7:43
for
S,
T being non
empty RelStr for
f being
set holds
( (
f is
Connection of
S,
T implies
f is
Connection of
S ~ ,
T ) & (
f is
Connection of
S ~ ,
T implies
f is
Connection of
S,
T ) & (
f is
Connection of
S,
T implies
f is
Connection of
S,
T ~ ) & (
f is
Connection of
S,
T ~ implies
f is
Connection of
S,
T ) & (
f is
Connection of
S,
T implies
f is
Connection of
S ~ ,
T ~ ) & (
f is
Connection of
S ~ ,
T ~ implies
f is
Connection of
S,
T ) )
theorem :: YELLOW_7:44
theorem Th45: :: YELLOW_7:45
theorem :: YELLOW_7:46
theorem Th47: :: YELLOW_7:47
theorem Th48: :: YELLOW_7:48
theorem Th49: :: YELLOW_7:49
theorem Th50: :: YELLOW_7:50
theorem :: YELLOW_7:51