:: 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
F being
Field for
a,
b being
Element of
suppf F for
c,
d being
Element of
(suppf F) \ {(ndf F)} holds
(odf F) . [((omf F) . [a,((revf F) . c)]),((omf F) . [b,((revf F) . d)])] = (omf F) . [((odf F) . [((omf F) . a,d),((omf F) . b,c)]),((revf F) . ((omf F) . c,d))]
definition
let F be
Field;
func osf c1 -> BinOp of
suppf a1 means :
Def1:
:: REALSET3:def 1
for
x,
y being
Element of
suppf F holds
it . x,
y = (odf F) . x,
((compf F) . y);
existence
ex b1 being BinOp of suppf F st
for x, y being Element of suppf F holds b1 . x,y = (odf F) . x,((compf F) . y)
uniqueness
for b1, b2 being BinOp of suppf F st ( for x, y being Element of suppf F holds b1 . x,y = (odf F) . x,((compf F) . y) ) & ( for x, y being Element of suppf F holds b2 . x,y = (odf F) . x,((compf F) . y) ) holds
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 F be
Field;
func ovf c1 -> Function of
[:(suppf a1),((suppf a1) \ {(ndf a1)}):],
suppf a1 means :
Def2:
:: REALSET3:def 2
for
x being
Element of
suppf F for
y being
Element of
(suppf F) \ {(ndf F)} holds
it . x,
y = (omf F) . x,
((revf F) . y);
existence
ex b1 being Function of [:(suppf F),((suppf F) \ {(ndf F)}):], suppf F st
for x being Element of suppf F
for y being Element of (suppf F) \ {(ndf F)} holds b1 . x,y = (omf F) . x,((revf F) . y)
uniqueness
for b1, b2 being Function of [:(suppf F),((suppf F) \ {(ndf F)}):], suppf F st ( for x being Element of suppf F
for y being Element of (suppf F) \ {(ndf F)} holds b1 . x,y = (omf F) . x,((revf F) . y) ) & ( for x being Element of suppf F
for y being Element of (suppf F) \ {(ndf F)} holds b2 . x,y = (omf F) . x,((revf F) . y) ) holds
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