:: NEWTON semantic presentation
theorem Th1: :: NEWTON:1
canceled;
theorem Th2: :: NEWTON:2
canceled;
theorem Th3: :: NEWTON:3
theorem Th4: :: NEWTON:4
canceled;
theorem Th5: :: NEWTON:5
for b
1 being
Nat holds
( b
1 >= 1 implies
Seg b
1 = ({1} \/ { b2 where B is Nat : ( 1 < b2 & b2 < b1 ) } ) \/ {b1} )
theorem Th6: :: NEWTON:6
theorem Th7: :: NEWTON:7
:: deftheorem Def1 defines |^ NEWTON:def 1 :
theorem Th8: :: NEWTON:8
canceled;
theorem Th9: :: NEWTON:9
theorem Th10: :: NEWTON:10
theorem Th11: :: NEWTON:11
theorem Th12: :: NEWTON:12
theorem Th13: :: NEWTON:13
theorem Th14: :: NEWTON:14
theorem Th15: :: NEWTON:15
theorem Th16: :: NEWTON:16
:: deftheorem Def2 defines ! NEWTON:def 2 :
theorem Th17: :: NEWTON:17
canceled;
theorem Th18: :: NEWTON:18
theorem Th19: :: NEWTON:19
theorem Th20: :: NEWTON:20
theorem Th21: :: NEWTON:21
theorem Th22: :: NEWTON:22
theorem Th23: :: NEWTON:23
theorem Th24: :: NEWTON:24
canceled;
theorem Th25: :: NEWTON:25
:: deftheorem Def3 defines choose NEWTON:def 3 :
theorem Th26: :: NEWTON:26
canceled;
theorem Th27: :: NEWTON:27
theorem Th28: :: NEWTON:28
canceled;
theorem Th29: :: NEWTON:29
theorem Th30: :: NEWTON:30
theorem Th31: :: NEWTON:31
theorem Th32: :: NEWTON:32
theorem Th33: :: NEWTON:33
theorem Th34: :: NEWTON:34
theorem Th35: :: NEWTON:35
theorem Th36: :: NEWTON:36
definition
let c
1, c
2 be
real number ;
let c
3 be
natural number ;
func c
1,c
2 In_Power c
3 -> FinSequence of
REAL means :
Def4:
:: NEWTON:def 4
(
len a
4 = a
3 + 1 & ( for b
1, b
2, b
3 being
natural number holds
( b
1 in dom a
4 & b
3 = b
1 - 1 & b
2 = a
3 - b
3 implies a
4 . b
1 = ((a3 choose b3) * (a1 |^ b2)) * (a2 |^ b3) ) ) );
existence
ex b1 being FinSequence of REAL st
( len b1 = c3 + 1 & ( for b2, b3, b4 being natural number holds
( b2 in dom b1 & b4 = b2 - 1 & b3 = c3 - b4 implies b1 . b2 = ((c3 choose b4) * (c1 |^ b3)) * (c2 |^ b4) ) ) )
uniqueness
for b1, b2 being FinSequence of REAL holds
( len b1 = c3 + 1 & ( for b3, b4, b5 being natural number holds
( b3 in dom b1 & b5 = b3 - 1 & b4 = c3 - b5 implies b1 . b3 = ((c3 choose b5) * (c1 |^ b4)) * (c2 |^ b5) ) ) & len b2 = c3 + 1 & ( for b3, b4, b5 being natural number holds
( b3 in dom b2 & b5 = b3 - 1 & b4 = c3 - b5 implies b2 . b3 = ((c3 choose b5) * (c1 |^ b4)) * (c2 |^ b5) ) ) implies b1 = b2 )
end;
:: deftheorem Def4 defines In_Power NEWTON:def 4 :
theorem Th37: :: NEWTON:37
canceled;
theorem Th38: :: NEWTON:38
theorem Th39: :: NEWTON:39
theorem Th40: :: NEWTON:40
theorem Th41: :: NEWTON:41
:: deftheorem Def5 defines Newton_Coeff NEWTON:def 5 :
theorem Th42: :: NEWTON:42
canceled;
theorem Th43: :: NEWTON:43
theorem Th44: :: NEWTON:44
theorem Th45: :: NEWTON:45
for b
1, b
2 being
Nat holds
( b
1 >= 1 implies b
2 * b
1 >= b
2 )
theorem Th46: :: NEWTON:46
for b
1, b
2, b
3 being
Nat holds
( b
1 >= 1 & b
2 >= b
3 * b
1 implies b
2 >= b
3 )
theorem Th47: :: NEWTON:47
theorem Th48: :: NEWTON:48
for b
1 being
Nat holds
not ( b
1 <> 0 & not
(b1 + 1) / b
1 > 1 )
theorem Th49: :: NEWTON:49
for b
1 being
Nat holds
b
1 / (b1 + 1) < 1
theorem Th50: :: NEWTON:50
for b
1 being
Nat holds b
1 ! >= b
1
theorem Th51: :: NEWTON:51
theorem Th52: :: NEWTON:52
theorem Th53: :: NEWTON:53
theorem Th54: :: NEWTON:54
theorem Th55: :: NEWTON:55
for b
1, b
2 being
Nat holds
not ( b
2 <> 1 & b
2 <> 0 & b
2 divides (b1 ! ) + 1 & not b
2 > b
1 )
theorem Th56: :: NEWTON:56
theorem Th57: :: NEWTON:57
theorem Th58: :: NEWTON:58
theorem Th59: :: NEWTON:59
for b
1 being
Nat holds b
1 lcm 1
= b
1
theorem Th60: :: NEWTON:60
theorem Th61: :: NEWTON:61
theorem Th62: :: NEWTON:62
theorem Th63: :: NEWTON:63
theorem Th64: :: NEWTON:64
for b
1 being
Nat holds b
1 hcf 1
= 1
theorem Th65: :: NEWTON:65
for b
1 being
Nat holds b
1 hcf 0
= b
1
theorem Th66: :: NEWTON:66
for b
1, b
2 being
Nat holds
(b1 hcf b2) lcm b
2 = b
2
theorem Th67: :: NEWTON:67
for b
1, b
2 being
Nat holds b
1 hcf (b1 lcm b2) = b
1
theorem Th68: :: NEWTON:68
theorem Th69: :: NEWTON:69
theorem Th70: :: NEWTON:70
theorem Th71: :: NEWTON:71
for b
1, b
2 being
Nat holds
not ( b
1 > 0 & not b
1 hcf b
2 > 0 )
theorem Th72: :: NEWTON:72
canceled;
theorem Th73: :: NEWTON:73
for b
1, b
2 being
Nat holds
not ( b
1 > 0 & b
2 > 0 & not b
1 lcm b
2 > 0 )
theorem Th74: :: NEWTON:74
theorem Th75: :: NEWTON:75
theorem Th76: :: NEWTON:76
theorem Th77: :: NEWTON:77
for b
1, b
2 being
Nat holds
( 0
< b
2 implies b
1 mod b
2 = b
1 - (b2 * (b1 div b2)) )
theorem Th78: :: NEWTON:78
theorem Th79: :: NEWTON:79
for b
1, b
2 being
Integer holds
not ( b
1 > 0 & not b
2 mod b
1 < b
1 )
theorem Th80: :: NEWTON:80
theorem Th81: :: NEWTON:81
for b
1, b
2 being
Nat holds
not ( not ( not b
1 > 0 & not b
2 > 0 ) & ( for b
3, b
4 being
Integer holds
not
(b3 * b1) + (b4 * b2) = b
1 hcf b
2 ) )
:: deftheorem Def6 defines SetPrimes NEWTON:def 6 :
:: deftheorem Def7 defines SetPrimenumber NEWTON:def 7 :
theorem Th82: :: NEWTON:82
theorem Th83: :: NEWTON:83
theorem Th84: :: NEWTON:84
theorem Th85: :: NEWTON:85
Lemma51:
for b1 being Nat holds
not ( not b1 = 0 & not b1 = 1 & not 2 <= b1 )
theorem Th86: :: NEWTON:86
theorem Th87: :: NEWTON:87
theorem Th88: :: NEWTON:88
theorem Th89: :: NEWTON:89
theorem Th90: :: NEWTON:90
theorem Th91: :: NEWTON:91
theorem Th92: :: NEWTON:92
for b
1 being
Nat holds
not b
1 in { b2 where B is Nat : ( b2 < b1 & b2 is prime ) }
theorem Th93: :: NEWTON:93
theorem Th94: :: NEWTON:94
for b
1, b
2 being
Prime holds
( b
1 < b
2 implies
{ b3 where B is Nat : ( b3 < b1 & b3 is prime ) } \/ {b1} c= { b3 where B is Nat : ( b3 < b2 & b3 is prime ) } )
theorem Th95: :: NEWTON:95
for b
1, b
2 being
Nat holds
not ( b
2 > b
1 & b
2 in { b3 where B is Nat : ( b3 < b1 & b3 is prime ) } )
:: deftheorem Def8 defines primenumber NEWTON:def 8 :
theorem Th96: :: NEWTON:96
theorem Th97: :: NEWTON:97
Lemma61:
for b1 being Nat holds
not ( b1 is prime & not b1 > 0 )
by INT_2:def 5;
theorem Th98: :: NEWTON:98
theorem Th99: :: NEWTON:99
for b
1, b
2 being
Nat holds
b
2 |^ b
1 is
Nat
theorem Th100: :: NEWTON:100
theorem Th101: :: NEWTON:101
theorem Th102: :: NEWTON:102