:: T_1TOPSP semantic presentation
theorem Th1: :: T_1TOPSP:1
canceled;
theorem Th2: :: T_1TOPSP:2
theorem Th3: :: T_1TOPSP:3
theorem Th4: :: T_1TOPSP:4
theorem Th5: :: T_1TOPSP:5
:: deftheorem Def1 defines EqClass T_1TOPSP:def 1 :
theorem Th6: :: T_1TOPSP:6
theorem Th7: :: T_1TOPSP:7
:: deftheorem Def2 defines Family-Class T_1TOPSP:def 2 :
Lemma42:
for X being set holds {} is Family-Class of X
:: deftheorem Def3 defines partition-membered T_1TOPSP:def 3 :
theorem Th8: :: T_1TOPSP:8
theorem Th9: :: T_1TOPSP:9
Lemma49:
for X being non empty set
for x being Element of X
for F being Part-Family of X
for A being set st A in { (EqClass x,S) where S is a_partition of X : S in F } holds
ex T being a_partition of X st
( T in F & A = EqClass x,T )
theorem Th10: :: T_1TOPSP:10
:: deftheorem Def4 defines Intersection T_1TOPSP:def 4 :
theorem Th11: :: T_1TOPSP:11
:: deftheorem Def5 defines Closed_Partitions T_1TOPSP:def 5 :
:: deftheorem Def6 defines T_1-reflex T_1TOPSP:def 6 :
theorem Th12: :: T_1TOPSP:12
:: deftheorem Def7 defines T_1-reflect T_1TOPSP:def 7 :
theorem Th13: :: T_1TOPSP:13
theorem Th14: :: T_1TOPSP:14
theorem Th15: :: T_1TOPSP:15
theorem Th16: :: T_1TOPSP:16
:: deftheorem Def8 defines T_1-reflex T_1TOPSP:def 8 :