:: SEQFUNC semantic presentation
:: deftheorem Def1 defines Functional_Sequence SEQFUNC:def 1 :
Lemma2:
for b1, b2 being set
for b3 being Function holds
( b3 is Functional_Sequence of b1,b2 iff ( dom b3 = NAT & ( for b4 being set holds
( b4 in NAT implies b3 . b4 is PartFunc of b1,b2 ) ) ) )
theorem Th1: :: SEQFUNC:1
Lemma3:
for b1, b2 being set
for b3, b4 being Functional_Sequence of b2,b1 holds
( ( for b5 being set holds
( b5 in NAT implies b3 . b5 = b4 . b5 ) ) implies b3 = b4 )
theorem Th2: :: SEQFUNC:2
:: deftheorem Def2 defines (#) SEQFUNC:def 2 :
definition
let c
1 be non
empty set ;
let c
2 be
Functional_Sequence of c
1,
REAL ;
func c
2 " -> Functional_Sequence of a
1,
REAL means :
Def3:
:: SEQFUNC:def 3
for b
1 being
Nat holds a
3 . b
1 = (a2 . b1) ^ ;
existence
ex b1 being Functional_Sequence of c1, REAL st
for b2 being Nat holds b1 . b2 = (c2 . b2) ^
uniqueness
for b1, b2 being Functional_Sequence of c1, REAL holds
( ( for b3 being Nat holds b1 . b3 = (c2 . b3) ^ ) & ( for b3 being Nat holds b2 . b3 = (c2 . b3) ^ ) implies b1 = b2 )
func - c
2 -> Functional_Sequence of a
1,
REAL means :
Def4:
:: SEQFUNC:def 4
for b
1 being
Nat holds a
3 . b
1 = - (a2 . b1);
existence
ex b1 being Functional_Sequence of c1, REAL st
for b2 being Nat holds b1 . b2 = - (c2 . b2)
uniqueness
for b1, b2 being Functional_Sequence of c1, REAL holds
( ( for b3 being Nat holds b1 . b3 = - (c2 . b3) ) & ( for b3 being Nat holds b2 . b3 = - (c2 . b3) ) implies b1 = b2 )
func abs c
2 -> Functional_Sequence of a
1,
REAL means :
Def5:
:: SEQFUNC:def 5
for b
1 being
Nat holds a
3 . b
1 = abs (a2 . b1);
existence
ex b1 being Functional_Sequence of c1, REAL st
for b2 being Nat holds b1 . b2 = abs (c2 . b2)
uniqueness
for b1, b2 being Functional_Sequence of c1, REAL holds
( ( for b3 being Nat holds b1 . b3 = abs (c2 . b3) ) & ( for b3 being Nat holds b2 . b3 = abs (c2 . b3) ) implies b1 = b2 )
end;
:: deftheorem Def3 defines " SEQFUNC:def 3 :
:: deftheorem Def4 defines - SEQFUNC:def 4 :
:: deftheorem Def5 defines abs SEQFUNC:def 5 :
:: deftheorem Def6 defines + SEQFUNC:def 6 :
:: deftheorem Def7 defines - SEQFUNC:def 7 :
:: deftheorem Def8 defines (#) SEQFUNC:def 8 :
:: deftheorem Def9 defines / SEQFUNC:def 9 :
theorem Th3: :: SEQFUNC:3
theorem Th4: :: SEQFUNC:4
theorem Th5: :: SEQFUNC:5
theorem Th6: :: SEQFUNC:6
theorem Th7: :: SEQFUNC:7
theorem Th8: :: SEQFUNC:8
theorem Th9: :: SEQFUNC:9
theorem Th10: :: SEQFUNC:10
theorem Th11: :: SEQFUNC:11
theorem Th12: :: SEQFUNC:12
theorem Th13: :: SEQFUNC:13
theorem Th14: :: SEQFUNC:14
theorem Th15: :: SEQFUNC:15
theorem Th16: :: SEQFUNC:16
theorem Th17: :: SEQFUNC:17
theorem Th18: :: SEQFUNC:18
theorem Th19: :: SEQFUNC:19
:: deftheorem Def10 defines common_on_dom SEQFUNC:def 10 :
:: deftheorem Def11 defines # SEQFUNC:def 11 :
:: deftheorem Def12 defines is_point_conv_on SEQFUNC:def 12 :
theorem Th20: :: SEQFUNC:20
theorem Th21: :: SEQFUNC:21
:: deftheorem Def13 defines is_unif_conv_on SEQFUNC:def 13 :
:: deftheorem Def14 defines lim SEQFUNC:def 14 :
theorem Th22: :: SEQFUNC:22
theorem Th23: :: SEQFUNC:23
theorem Th24: :: SEQFUNC:24
theorem Th25: :: SEQFUNC:25
theorem Th26: :: SEQFUNC:26
theorem Th27: :: SEQFUNC:27
theorem Th28: :: SEQFUNC:28
theorem Th29: :: SEQFUNC:29
theorem Th30: :: SEQFUNC:30
theorem Th31: :: SEQFUNC:31
theorem Th32: :: SEQFUNC:32
theorem Th33: :: SEQFUNC:33
theorem Th34: :: SEQFUNC:34
theorem Th35: :: SEQFUNC:35
theorem Th36: :: SEQFUNC:36
theorem Th37: :: SEQFUNC:37
theorem Th38: :: SEQFUNC:38
theorem Th39: :: SEQFUNC:39
theorem Th40: :: SEQFUNC:40
theorem Th41: :: SEQFUNC:41
for b
1 being non
empty set for b
2, b
3 being
Functional_Sequence of b
1,
REAL for b
4 being
set holds
( b
2 is_point_conv_on b
4 & b
3 is_point_conv_on b
4 implies ( b
2 + b
3 is_point_conv_on b
4 &
lim (b2 + b3),b
4 = (lim b2,b4) + (lim b3,b4) & b
2 - b
3 is_point_conv_on b
4 &
lim (b2 - b3),b
4 = (lim b2,b4) - (lim b3,b4) & b
2 (#) b
3 is_point_conv_on b
4 &
lim (b2 (#) b3),b
4 = (lim b2,b4) (#) (lim b3,b4) ) )
theorem Th42: :: SEQFUNC:42
theorem Th43: :: SEQFUNC:43
theorem Th44: :: SEQFUNC:44
theorem Th45: :: SEQFUNC:45