:: REALSET3 semantic presentation
theorem Th1: :: REALSET3:1
theorem Th2: :: REALSET3:2
theorem Th3: :: REALSET3:3
theorem Th4: :: REALSET3:4
theorem Th5: :: REALSET3:5
theorem Th6: :: REALSET3:6
theorem Th7: :: REALSET3:7
theorem Th8: :: REALSET3:8
theorem Th9: :: REALSET3:9
theorem Th10: :: REALSET3:10
theorem Th11: :: REALSET3:11
for b
1 being
Fieldfor b
2, b
3 being
Element of
suppf b
1for b
4, b
5 being
Element of
(suppf b1) \ {(ndf b1)} holds
(odf b1) . [((omf b1) . [b2,((revf b1) . b4)]),((omf b1) . [b3,((revf b1) . b5)])] = (omf b1) . [((odf b1) . [((omf b1) . b2,b5),((omf b1) . b3,b4)]),((revf b1) . ((omf b1) . b4,b5))]
definition
let c
1 be
Field;
func osf c
1 -> BinOp of
suppf a
1 means :
Def1:
:: REALSET3:def 1
for b
1, b
2 being
Element of
suppf a
1 holds a
2 . b
1,b
2 = (odf a1) . b
1,
((compf a1) . b2);
existence
ex b1 being BinOp of suppf c1 st
for b2, b3 being Element of suppf c1 holds b1 . b2,b3 = (odf c1) . b2,((compf c1) . b3)
uniqueness
for b1, b2 being BinOp of suppf c1 holds
( ( for b3, b4 being Element of suppf c1 holds b1 . b3,b4 = (odf c1) . b3,((compf c1) . b4) ) & ( for b3, b4 being Element of suppf c1 holds b2 . b3,b4 = (odf c1) . b3,((compf c1) . b4) ) implies b1 = b2 )
end;
:: deftheorem Def1 defines osf REALSET3:def 1 :
theorem Th12: :: REALSET3:12
canceled;
theorem Th13: :: REALSET3:13
canceled;
theorem Th14: :: REALSET3:14
theorem Th15: :: REALSET3:15
theorem Th16: :: REALSET3:16
theorem Th17: :: REALSET3:17
theorem Th18: :: REALSET3:18
theorem Th19: :: REALSET3:19
theorem Th20: :: REALSET3:20
theorem Th21: :: REALSET3:21
theorem Th22: :: REALSET3:22
theorem Th23: :: REALSET3:23
theorem Th24: :: REALSET3:24
theorem Th25: :: REALSET3:25
theorem Th26: :: REALSET3:26
definition
let c
1 be
Field;
func ovf c
1 -> Function of
[:(suppf a1),((suppf a1) \ {(ndf a1)}):],
suppf a
1 means :
Def2:
:: REALSET3:def 2
for b
1 being
Element of
suppf a
1for b
2 being
Element of
(suppf a1) \ {(ndf a1)} holds a
2 . b
1,b
2 = (omf a1) . b
1,
((revf a1) . b2);
existence
ex b1 being Function of [:(suppf c1),((suppf c1) \ {(ndf c1)}):], suppf c1 st
for b2 being Element of suppf c1
for b3 being Element of (suppf c1) \ {(ndf c1)} holds b1 . b2,b3 = (omf c1) . b2,((revf c1) . b3)
uniqueness
for b1, b2 being Function of [:(suppf c1),((suppf c1) \ {(ndf c1)}):], suppf c1 holds
( ( for b3 being Element of suppf c1
for b4 being Element of (suppf c1) \ {(ndf c1)} holds b1 . b3,b4 = (omf c1) . b3,((revf c1) . b4) ) & ( for b3 being Element of suppf c1
for b4 being Element of (suppf c1) \ {(ndf c1)} holds b2 . b3,b4 = (omf c1) . b3,((revf c1) . b4) ) implies b1 = b2 )
end;
:: deftheorem Def2 defines ovf REALSET3:def 2 :
theorem Th27: :: REALSET3:27
canceled;
theorem Th28: :: REALSET3:28
canceled;
theorem Th29: :: REALSET3:29
theorem Th30: :: REALSET3:30
theorem Th31: :: REALSET3:31
theorem Th32: :: REALSET3:32
theorem Th33: :: REALSET3:33
canceled;
theorem Th34: :: REALSET3:34
canceled;
theorem Th35: :: REALSET3:35
theorem Th36: :: REALSET3:36
theorem Th37: :: REALSET3:37
theorem Th38: :: REALSET3:38
theorem Th39: :: REALSET3:39
theorem Th40: :: REALSET3:40
theorem Th41: :: REALSET3:41
theorem Th42: :: REALSET3:42
theorem Th43: :: REALSET3:43