:: MESFUNC2 semantic presentation
:: deftheorem Def1 defines is_finite MESFUNC2:def 1 :
theorem Th1: :: MESFUNC2:1
theorem Th2: :: MESFUNC2:2
theorem Th3: :: MESFUNC2:3
theorem Th4: :: MESFUNC2:4
theorem Th5: :: MESFUNC2:5
theorem Th6: :: MESFUNC2:6
theorem Th7: :: MESFUNC2:7
theorem Th8: :: MESFUNC2:8
canceled;
theorem Th9: :: MESFUNC2:9
theorem Th10: :: MESFUNC2:10
theorem Th11: :: MESFUNC2:11
theorem Th12: :: MESFUNC2:12
theorem Th13: :: MESFUNC2:13
definition
let c
1 be non
empty set ;
let c
2 be
PartFunc of c
1,
ExtREAL ;
deffunc H
1(
Element of c
1)
-> Element of
ExtREAL =
max (c2 . a1),
0. ;
func max+ c
2 -> PartFunc of a
1,
ExtREAL means :
Def2:
:: MESFUNC2:def 2
(
dom a
3 = dom a
2 & ( for b
1 being
Element of a
1 holds
( b
1 in dom a
3 implies a
3 . b
1 = max (a2 . b1),
0. ) ) );
existence
ex b1 being PartFunc of c1, ExtREAL st
( dom b1 = dom c2 & ( for b2 being Element of c1 holds
( b2 in dom b1 implies b1 . b2 = max (c2 . b2),0. ) ) )
uniqueness
for b1, b2 being PartFunc of c1, ExtREAL holds
( dom b1 = dom c2 & ( for b3 being Element of c1 holds
( b3 in dom b1 implies b1 . b3 = max (c2 . b3),0. ) ) & dom b2 = dom c2 & ( for b3 being Element of c1 holds
( b3 in dom b2 implies b2 . b3 = max (c2 . b3),0. ) ) implies b1 = b2 )
deffunc H
2(
Element of c
1)
-> Element of
ExtREAL =
max (- (c2 . a1)),
0. ;
func max- c
2 -> PartFunc of a
1,
ExtREAL means :
Def3:
:: MESFUNC2:def 3
(
dom a
3 = dom a
2 & ( for b
1 being
Element of a
1 holds
( b
1 in dom a
3 implies a
3 . b
1 = max (- (a2 . b1)),
0. ) ) );
existence
ex b1 being PartFunc of c1, ExtREAL st
( dom b1 = dom c2 & ( for b2 being Element of c1 holds
( b2 in dom b1 implies b1 . b2 = max (- (c2 . b2)),0. ) ) )
uniqueness
for b1, b2 being PartFunc of c1, ExtREAL holds
( dom b1 = dom c2 & ( for b3 being Element of c1 holds
( b3 in dom b1 implies b1 . b3 = max (- (c2 . b3)),0. ) ) & dom b2 = dom c2 & ( for b3 being Element of c1 holds
( b3 in dom b2 implies b2 . b3 = max (- (c2 . b3)),0. ) ) implies b1 = b2 )
end;
:: deftheorem Def2 defines max+ MESFUNC2:def 2 :
:: deftheorem Def3 defines max- MESFUNC2:def 3 :
theorem Th14: :: MESFUNC2:14
theorem Th15: :: MESFUNC2:15
theorem Th16: :: MESFUNC2:16
theorem Th17: :: MESFUNC2:17
theorem Th18: :: MESFUNC2:18
theorem Th19: :: MESFUNC2:19
theorem Th20: :: MESFUNC2:20
theorem Th21: :: MESFUNC2:21
theorem Th22: :: MESFUNC2:22
theorem Th23: :: MESFUNC2:23
theorem Th24: :: MESFUNC2:24
theorem Th25: :: MESFUNC2:25
theorem Th26: :: MESFUNC2:26
theorem Th27: :: MESFUNC2:27
theorem Th28: :: MESFUNC2:28
theorem Th29: :: MESFUNC2:29
theorem Th30: :: MESFUNC2:30
theorem Th31: :: MESFUNC2:31
theorem Th32: :: MESFUNC2:32
theorem Th33: :: MESFUNC2:33
theorem Th34: :: MESFUNC2:34
:: deftheorem Def4 MESFUNC2:def 4 :
canceled;
:: deftheorem Def5 defines is_simple_func_in MESFUNC2:def 5 :
theorem Th35: :: MESFUNC2:35
theorem Th36: :: MESFUNC2:36
theorem Th37: :: MESFUNC2:37