:: RFINSEQ semantic presentation
:: deftheorem Def1 defines are_fiberwise_equipotent RFINSEQ:def 1 :
Lemma2:
for b1 being Function
for b2 being set holds
( not b2 in rng b1 implies b1 " {b2} = {} )
theorem Th1: :: RFINSEQ:1
theorem Th2: :: RFINSEQ:2
theorem Th3: :: RFINSEQ:3
theorem Th4: :: RFINSEQ:4
theorem Th5: :: RFINSEQ:5
theorem Th6: :: RFINSEQ:6
theorem Th7: :: RFINSEQ:7
theorem Th8: :: RFINSEQ:8
canceled;
theorem Th9: :: RFINSEQ:9
theorem Th10: :: RFINSEQ:10
theorem Th11: :: RFINSEQ:11
theorem Th12: :: RFINSEQ:12
canceled;
theorem Th13: :: RFINSEQ:13
theorem Th14: :: RFINSEQ:14
theorem Th15: :: RFINSEQ:15
theorem Th16: :: RFINSEQ:16
theorem Th17: :: RFINSEQ:17
defpred S1[ Nat] means for b1 being finite set
for b2 being Function holds
not ( card (dom (b2 | b1)) = a1 & ( for b3 being FinSequence holds
not b2 | b1,b3 are_fiberwise_equipotent ) );
Lemma10:
S1[0]
Lemma11:
for b1 being Nat holds
( S1[b1] implies S1[b1 + 1] )
theorem Th18: :: RFINSEQ:18
:: deftheorem Def2 defines /^ RFINSEQ:def 2 :
Lemma13:
for b1 being Nat
for b2 being non empty set
for b3 being FinSequence of b2 holds
( len b3 <= b1 implies b3 | b1 = b3 )
theorem Th19: :: RFINSEQ:19
theorem Th20: :: RFINSEQ:20
theorem Th21: :: RFINSEQ:21
theorem Th22: :: RFINSEQ:22
:: deftheorem Def3 defines MIM RFINSEQ:def 3 :
theorem Th23: :: RFINSEQ:23
theorem Th24: :: RFINSEQ:24
theorem Th25: :: RFINSEQ:25
theorem Th26: :: RFINSEQ:26
theorem Th27: :: RFINSEQ:27
theorem Th28: :: RFINSEQ:28
theorem Th29: :: RFINSEQ:29
theorem Th30: :: RFINSEQ:30
:: deftheorem Def4 defines non-increasing RFINSEQ:def 4 :
theorem Th31: :: RFINSEQ:31
theorem Th32: :: RFINSEQ:32
theorem Th33: :: RFINSEQ:33
theorem Th34: :: RFINSEQ:34
Lemma27:
for b1, b2 being non-increasing FinSequence of REAL
for b3 being Nat holds
( len b1 = b3 + 1 & len b1 = len b2 & b1,b2 are_fiberwise_equipotent implies ( b1 . (len b1) = b2 . (len b2) & b1 | b3,b2 | b3 are_fiberwise_equipotent ) )
defpred S2[ Nat] means for b1 being FinSequence of REAL holds
not ( a1 = len b1 & ( for b2 being non-increasing FinSequence of REAL holds
not b1,b2 are_fiberwise_equipotent ) );
Lemma28:
S2[0]
Lemma29:
for b1 being Nat holds
( S2[b1] implies S2[b1 + 1] )
theorem Th35: :: RFINSEQ:35
Lemma30:
for b1 being Nat
for b2, b3 being non-increasing FinSequence of REAL holds
( b1 = len b2 & b2,b3 are_fiberwise_equipotent implies b2 = b3 )
theorem Th36: :: RFINSEQ:36
theorem Th37: :: RFINSEQ:37
theorem Th38: :: RFINSEQ:38
theorem Th39: :: RFINSEQ:39