:: EQREL_1 semantic presentation
:: deftheorem Def1 defines nabla EQREL_1:def 1 :
Lemma36:
for i, j being Element of NAT st i < j holds
j - i is Element of NAT
by NAT_1:61;
theorem Th1: :: EQREL_1:1
canceled;
theorem Th2: :: EQREL_1:2
canceled;
theorem Th3: :: EQREL_1:3
canceled;
theorem Th4: :: EQREL_1:4
theorem Th5: :: EQREL_1:5
canceled;
theorem Th6: :: EQREL_1:6
theorem Th7: :: EQREL_1:7
Lemma43:
for x, y, X being set
for R being Relation of X st [x,y] in R holds
( x in X & y in X )
theorem Th8: :: EQREL_1:8
canceled;
theorem Th9: :: EQREL_1:9
canceled;
theorem Th10: :: EQREL_1:10
canceled;
theorem Th11: :: EQREL_1:11
theorem Th12: :: EQREL_1:12
theorem Th13: :: EQREL_1:13
theorem Th14: :: EQREL_1:14
theorem Th15: :: EQREL_1:15
canceled;
theorem Th16: :: EQREL_1:16
theorem Th17: :: EQREL_1:17
theorem Th18: :: EQREL_1:18
theorem Th19: :: EQREL_1:19
theorem Th20: :: EQREL_1:20
:: deftheorem Def2 EQREL_1:def 2 :
canceled;
:: deftheorem Def3 defines "\/" EQREL_1:def 3 :
theorem Th21: :: EQREL_1:21
canceled;
theorem Th22: :: EQREL_1:22
theorem Th23: :: EQREL_1:23
theorem Th24: :: EQREL_1:24
theorem Th25: :: EQREL_1:25
scheme :: EQREL_1:sch 65
s65{
F1()
-> set ,
P1[
set ,
set ] } :
provided
E38:
for
x being
set st
x in F1() holds
P1[
x,
x]
and E39:
for
x,
y being
set st
P1[
x,
y] holds
P1[
y,
x]
and E48:
for
x,
y,
z being
set st
P1[
x,
y] &
P1[
y,
z] holds
P1[
x,
z]
:: deftheorem Def4 defines Class EQREL_1:def 4 :
theorem Th26: :: EQREL_1:26
theorem Th27: :: EQREL_1:27
theorem Th28: :: EQREL_1:28
theorem Th29: :: EQREL_1:29
theorem Th30: :: EQREL_1:30
Lemma68:
for X, y being set
for EqR being Equivalence_Relation of X
for x being set st x in X holds
( [x,y] in EqR iff Class EqR,x = Class EqR,y )
theorem Th31: :: EQREL_1:31
theorem Th32: :: EQREL_1:32
theorem Th33: :: EQREL_1:33
theorem Th34: :: EQREL_1:34
theorem Th35: :: EQREL_1:35
theorem Th36: :: EQREL_1:36
theorem Th37: :: EQREL_1:37
theorem Th38: :: EQREL_1:38
:: deftheorem Def5 defines Class EQREL_1:def 5 :
theorem Th39: :: EQREL_1:39
canceled;
theorem Th40: :: EQREL_1:40
:: deftheorem Def6 defines a_partition EQREL_1:def 6 :
theorem Th41: :: EQREL_1:41
canceled;
theorem Th42: :: EQREL_1:42
theorem Th43: :: EQREL_1:43
theorem Th44: :: EQREL_1:44
theorem Th45: :: EQREL_1:45
:: deftheorem Def7 defines SmallestPartition EQREL_1:def 7 :
theorem Th46: :: EQREL_1:46