:: REALSET1 semantic presentation
theorem Th1: :: REALSET1:1
theorem Th2: :: REALSET1:2
for b
1 being
set for b
2 being
BinOp of b
1 holds
ex b
3 being
Subset of b
1 st
for b
4 being
set holds
( b
4 in [:b3,b3:] implies b
2 . b
4 in b
3 )
:: deftheorem Def1 defines is_in REALSET1:def 1 :
for b
1 being
set for b
2 being
BinOp of b
1for b
3 being
Subset of b
1 holds
( b
2 is_in b
3 iff for b
4 being
set holds
( b
4 in [:b3,b3:] implies b
2 . b
4 in b
3 ) );
:: deftheorem Def2 defines Preserv REALSET1:def 2 :
for b
1 being
set for b
2 being
BinOp of b
1for b
3 being
Subset of b
1 holds
( b
3 is
Preserv of b
2 iff for b
4 being
set holds
( b
4 in [:b3,b3:] implies b
2 . b
4 in b
3 ) );
:: deftheorem Def3 defines || REALSET1:def 3 :
theorem Th3: :: REALSET1:3
:: deftheorem Def4 defines trivial REALSET1:def 4 :
for b
1 being
set holds
( b
1 is
trivial iff not ( not b
1 is
empty & ( for b
2 being
set holds
not b
1 = {b2} ) ) );
theorem Th4: :: REALSET1:4
theorem Th5: :: REALSET1:5
theorem Th6: :: REALSET1:6
theorem Th7: :: REALSET1:7
:: deftheorem Def5 defines is_Bin_Op_Preserv REALSET1:def 5 :
theorem Th8: :: REALSET1:8
for b
1 being
set for b
2 being
Subset of b
1 holds
ex b
3 being
BinOp of b
1 st
for b
4 being
set holds
( b
4 in [:b2,b2:] implies b
3 . b
4 in b
2 )
:: deftheorem Def6 defines Presv REALSET1:def 6 :
for b
1 being
set for b
2 being
Subset of b
1for b
3 being
BinOp of b
1 holds
( b
3 is
Presv of b
1,b
2 iff for b
4 being
set holds
( b
4 in [:b2,b2:] implies b
3 . b
4 in b
2 ) );
theorem Th9: :: REALSET1:9
:: deftheorem Def7 defines ||| REALSET1:def 7 :
theorem Th10: :: REALSET1:10
:: deftheorem Def8 defines DnT REALSET1:def 8 :
theorem Th11: :: REALSET1:11
:: deftheorem Def9 defines ! REALSET1:def 9 :
:: deftheorem Def10 defines OnePoint REALSET1:def 10 :
theorem Th12: :: REALSET1:12
theorem Th13: :: REALSET1:13
theorem Th14: :: REALSET1:14
for b
1 being
set holds
( not ( not b
1 is
trivial & ( for b
2, b
3 being
set holds
not ( b
2 in b
1 & b
3 in b
1 & b
2 <> b
3 ) ) ) & not ( ex b
2, b
3 being
set st
( b
2 in b
1 & b
3 in b
1 & b
2 <> b
3 ) & b
1 is
trivial ) )
theorem Th15: :: REALSET1:15
for b
1 being
set for b
2 being
Subset of b
1 holds
( not ( not b
2 is
trivial & ( for b
3, b
4 being
Element of b
1 holds
not ( b
3 in b
2 & b
4 in b
2 & b
3 <> b
4 ) ) ) & not ( ex b
3, b
4 being
Element of b
1 st
( b
3 in b
2 & b
4 in b
2 & b
3 <> b
4 ) & b
2 is
trivial ) )