:: Integer and Rational Exponents
:: by Konrad Raczkowski
::
:: Received September 21, 1990
:: Copyright (c) 1990 Association of Mizar Users


begin

registration
let i be Integer;
cluster |.i.| -> natural ;
coherence
abs i is natural ;
end;

theorem :: PREPOWER:1
canceled;

theorem Th2: :: PREPOWER:2
for a being real number
for s1 being Real_Sequence st s1 is convergent & ( for n being Element of NAT holds s1 . n >= a ) holds
lim s1 >= a
proof end;

theorem Th3: :: PREPOWER:3
for a being real number
for s1 being Real_Sequence st s1 is convergent & ( for n being Element of NAT holds s1 . n <= a ) holds
lim s1 <= a
proof end;

definition
let a be real number ;
func a GeoSeq -> Real_Sequence means :Def1: :: PREPOWER:def 1
 for m being Element of NAT holds it . m = a |^ m;
existence
ex b1 being Real_Sequence st
for m being Element of NAT holds b1 . m = a |^ m
proof end;
uniqueness
for b1, b2 being Real_Sequence st ( for m being Element of NAT holds b1 . m = a |^ m ) & ( for m being Element of NAT holds b2 . m = a |^ m ) holds
b1 = b2
proof end;
end;

:: deftheorem Def1 defines GeoSeq PREPOWER:def 1 :
for a being real number
for b2 being Real_Sequence holds
( b2 = a GeoSeq iff for m being Element of NAT holds b2 . m = a |^ m );

theorem Th4: :: PREPOWER:4
for a being real number
for s1 being Real_Sequence holds
( s1 = a GeoSeq iff ( s1 . 0 = 1 & ( for m being Element of NAT holds s1 . (m + 1) = (s1 . m) * a ) ) )
proof end;

theorem :: PREPOWER:5
for a being real number st a <> 0 holds
for m being Element of NAT holds (a GeoSeq) . m <> 0
proof end;

theorem :: PREPOWER:6
canceled;

theorem :: PREPOWER:7
canceled;

theorem :: PREPOWER:8
canceled;

theorem :: PREPOWER:9
canceled;

theorem :: PREPOWER:10
canceled;

theorem :: PREPOWER:11
canceled;

theorem Th12: :: PREPOWER:12
for a being real number
for n being natural number st 0 <> a holds
0 <> a |^ n
proof end;

theorem Th13: :: PREPOWER:13
for a being real number
for n being natural number st 0 < a holds
0 < a |^ n
proof end;

theorem Th14: :: PREPOWER:14
for a being real number
for n being natural number holds (1 / a) |^ n = 1 / (a |^ n)
proof end;

theorem :: PREPOWER:15
for b, a being real number
for n being natural number holds (b / a) |^ n = (b |^ n) / (a |^ n)
proof end;

theorem :: PREPOWER:16
canceled;

theorem Th17: :: PREPOWER:17
for a, b being real number
for n being natural number st 0 < a & a <= b holds
a |^ n <= b |^ n
proof end;

Lm1: for a, b being real number
for n being natural number st 0 < a & a < b & 1 <= n holds
a |^ n < b |^ n
proof end;

theorem Th18: :: PREPOWER:18
for a, b being real number
for n being natural number st 0 <= a & a < b & 1 <= n holds
a |^ n < b |^ n
proof end;

theorem Th19: :: PREPOWER:19
for a being real number
for n being natural number st a >= 1 holds
a |^ n >= 1
proof end;

theorem Th20: :: PREPOWER:20
for a being real number
for n being natural number st 1 <= a & 1 <= n holds
a <= a |^ n
proof end;

theorem :: PREPOWER:21
for a being real number
for n being natural number st 1 < a & 2 <= n holds
a < a |^ n
proof end;

theorem Th22: :: PREPOWER:22
for a being real number
for n being natural number st 0 < a & a <= 1 & 1 <= n holds
a |^ n <= a
proof end;

theorem :: PREPOWER:23
for a being real number
for n being natural number st 0 < a & a < 1 & 2 <= n holds
a |^ n < a
proof end;

theorem Th24: :: PREPOWER:24
for a being real number
for n being natural number st - 1 < a holds
(1 + a) |^ n >= 1 + (n * a)
proof end;

theorem Th25: :: PREPOWER:25
for a being real number
for n being natural number st 0 < a & a < 1 holds
(1 + a) |^ n <= 1 + ((3 |^ n) * a)
proof end;

theorem Th26: :: PREPOWER:26
for m being Element of NAT
for s1, s2 being Real_Sequence st s1 is convergent & ( for n being Element of NAT holds s2 . n = (s1 . n) |^ m ) holds
( s2 is convergent & lim s2 = (lim s1) |^ m )
proof end;

definition
let n be natural number ;
let a be real number ;
assume A1: 1 <= n ;
canceled;
func n -Root a -> real number means :Def3: :: PREPOWER:def 3
( it |^ n = a & it > 0 ) if a > 0
it = 0 if a = 0
;
consistency
for b1 being real number st a > 0 & a = 0 holds
( ( b1 |^ n = a & b1 > 0 ) iff b1 = 0 ) ;
existence
( ( a > 0 implies ex b1 being real number st
( b1 |^ n = a & b1 > 0 ) ) & ( a = 0 implies ex b1 being real number st b1 = 0 ) )
proof end;
uniqueness
for b1, b2 being real number holds
( ( a > 0 & b1 |^ n = a & b1 > 0 & b2 |^ n = a & b2 > 0 implies b1 = b2 ) & ( a = 0 & b1 = 0 & b2 = 0 implies b1 = b2 ) )
proof end;
end;

:: deftheorem PREPOWER:def 2 :
canceled;

:: deftheorem Def3 defines -Root PREPOWER:def 3 :
for n being natural number
for a being real number st 1 <= n holds
for b3 being real number holds
( ( a > 0 implies ( b3 = n -Root a iff ( b3 |^ n = a & b3 > 0 ) ) ) & ( a = 0 implies ( b3 = n -Root a iff b3 = 0 ) ) );

definition
let n be Element of NAT ;
let a be Real;
:: original: -Root
redefine func n -Root a -> Real;
coherence
n -Root a is Real by XREAL_0:def 1;
end;

Lm2: for a being real number
for n being Nat st a > 0 & n >= 1 holds
( (n -Root a) |^ n = a & n -Root (a |^ n) = a )
proof end;

theorem :: PREPOWER:27
canceled;

theorem Th28: :: PREPOWER:28
for a being real number
for n being Nat st a >= 0 & n >= 1 holds
( (n -Root a) |^ n = a & n -Root (a |^ n) = a )
proof end;

theorem Th29: :: PREPOWER:29
for n being Element of NAT st n >= 1 holds
n -Root 1 = 1
proof end;

theorem Th30: :: PREPOWER:30
for a being real number st a >= 0 holds
1 -Root a = a
proof end;

theorem Th31: :: PREPOWER:31
for a, b being real number
for n being Element of NAT st a >= 0 & b >= 0 & n >= 1 holds
n -Root (a * b) = (n -Root a) * (n -Root b)
proof end;

theorem Th32: :: PREPOWER:32
for a being real number
for n being Element of NAT st a > 0 & n >= 1 holds
n -Root (1 / a) = 1 / (n -Root a)
proof end;

theorem :: PREPOWER:33
for a, b being real number
for n being Element of NAT st a >= 0 & b > 0 & n >= 1 holds
n -Root (a / b) = (n -Root a) / (n -Root b)
proof end;

theorem Th34: :: PREPOWER:34
for a being real number
for n, m being Element of NAT st a >= 0 & n >= 1 & m >= 1 holds
n -Root (m -Root a) = (n * m) -Root a
proof end;

theorem Th35: :: PREPOWER:35
for a being real number
for n, m being Element of NAT st a >= 0 & n >= 1 & m >= 1 holds
(n -Root a) * (m -Root a) = (n * m) -Root (a |^ (n + m))
proof end;

theorem Th36: :: PREPOWER:36
for a, b being real number
for n being Element of NAT st 0 <= a & a <= b & n >= 1 holds
n -Root a <= n -Root b
proof end;

theorem Th37: :: PREPOWER:37
for a, b being real number
for n being Element of NAT st a >= 0 & a < b & n >= 1 holds
n -Root a < n -Root b
proof end;

theorem Th38: :: PREPOWER:38
for a being real number
for n being Element of NAT st a >= 1 & n >= 1 holds
( n -Root a >= 1 & a >= n -Root a )
proof end;

theorem Th39: :: PREPOWER:39
for a being real number
for n being Element of NAT st 0 <= a & a < 1 & n >= 1 holds
( a <= n -Root a & n -Root a < 1 )
proof end;

theorem Th40: :: PREPOWER:40
for a being real number
for n being Element of NAT st a > 0 & n >= 1 holds
(n -Root a) - 1 <= (a - 1) / n
proof end;

theorem :: PREPOWER:41
for a being real number st a >= 0 holds
2 -Root a = sqrt a
proof end;

Lm3: for s being Real_Sequence
for a being real number st a >= 1 & ( for n being Element of NAT st n >= 1 holds
s . n = n -Root a ) holds
( s is convergent & lim s = 1 )
proof end;

theorem :: PREPOWER:42
for a being real number
for s being Real_Sequence st a > 0 & ( for n being Element of NAT st n >= 1 holds
s . n = n -Root a ) holds
( s is convergent & lim s = 1 )
proof end;

definition
let a be real number ;
let k be Integer;
func a #Z k -> set equals :Def4: :: PREPOWER:def 4
a |^ (abs k) if k >= 0
(a |^ (abs k)) " if k < 0
;
consistency
for b1 being set st k >= 0 & k < 0 holds
( b1 = a |^ (abs k) iff b1 = (a |^ (abs k)) " ) ;
coherence
( ( k >= 0 implies a |^ (abs k) is set ) & ( k < 0 implies (a |^ (abs k)) " is set ) ) ;
end;

:: deftheorem Def4 defines #Z PREPOWER:def 4 :
for a being real number
for k being Integer holds
( ( k >= 0 implies a #Z k = a |^ (abs k) ) & ( k < 0 implies a #Z k = (a |^ (abs k)) " ) );

registration
let a be real number ;
let k be Integer;
cluster a #Z k -> real ;
coherence
a #Z k is real
proof end;
end;

definition
let a be Real;
let k be Integer;
:: original: #Z
redefine func a #Z k -> Real;
coherence
a #Z k is Real by XREAL_0:def 1;
end;

theorem :: PREPOWER:43
canceled;

theorem Th44: :: PREPOWER:44
for a being real number holds a #Z 0 = 1
proof end;

theorem Th45: :: PREPOWER:45
for a being real number holds a #Z 1 = a
proof end;

theorem Th46: :: PREPOWER:46
for a being real number
for n being natural number holds a #Z n = a |^ n
proof end;

Lm4: 1 " = 1
;

theorem Th47: :: PREPOWER:47
for k being Integer holds 1 #Z k = 1
proof end;

theorem :: PREPOWER:48
for a being real number
for k being Integer st a <> 0 holds
a #Z k <> 0
proof end;

theorem Th49: :: PREPOWER:49
for a being real number
for k being Integer st a > 0 holds
a #Z k > 0
proof end;

theorem Th50: :: PREPOWER:50
for a, b being real number
for k being Integer holds (a * b) #Z k = (a #Z k) * (b #Z k)
proof end;

theorem Th51: :: PREPOWER:51
for a being real number
for k being Integer holds a #Z (- k) = 1 / (a #Z k)
proof end;

theorem Th52: :: PREPOWER:52
for a being real number
for k being Integer holds (1 / a) #Z k = 1 / (a #Z k)
proof end;

theorem Th53: :: PREPOWER:53
for a being real number
for m, n being natural number st a <> 0 holds
a #Z (m - n) = (a |^ m) / (a |^ n)
proof end;

theorem Th54: :: PREPOWER:54
for a being real number
for k, l being Integer st a <> 0 holds
a #Z (k + l) = (a #Z k) * (a #Z l)
proof end;

theorem Th55: :: PREPOWER:55
for a being real number
for k, l being Integer holds (a #Z k) #Z l = a #Z (k * l)
proof end;

theorem Th56: :: PREPOWER:56
for a being real number
for n being Element of NAT
for k being Integer st a > 0 & n >= 1 holds
(n -Root a) #Z k = n -Root (a #Z k)
proof end;

definition
let a be real number ;
let p be Rational;
func a #Q p -> set equals :: PREPOWER:def 5
(denominator p) -Root (a #Z (numerator p));
coherence
(denominator p) -Root (a #Z (numerator p)) is set ;
end;

:: deftheorem defines #Q PREPOWER:def 5 :
for a being real number
for p being Rational holds a #Q p = (denominator p) -Root (a #Z (numerator p));

registration
let a be real number ;
let p be Rational;
cluster a #Q p -> real ;
coherence
a #Q p is real ;
end;

definition
let a be Real;
let p be Rational;
:: original: #Q
redefine func a #Q p -> Real;
coherence
a #Q p is Real by XREAL_0:def 1;
end;

theorem :: PREPOWER:57
canceled;

theorem Th58: :: PREPOWER:58
for a being real number
for p being Rational st p = 0 holds
a #Q p = 1
proof end;

theorem Th59: :: PREPOWER:59
for a being real number
for p being Rational st a > 0 & p = 1 holds
a #Q p = a
proof end;

Lm5: for a being real number
for k being Integer st a >= 0 holds
a #Z k >= 0
proof end;

theorem Th60: :: PREPOWER:60
for a being real number
for n being Nat st 0 <= a holds
a #Q n = a |^ n
proof end;

theorem Th61: :: PREPOWER:61
for a being real number
for p being Rational
for n being Nat st n >= 1 & p = n " holds
a #Q p = n -Root a
proof end;

theorem Th62: :: PREPOWER:62
for p being Rational holds 1 #Q p = 1
proof end;

theorem Th63: :: PREPOWER:63
for a being real number
for p being Rational st a > 0 holds
a #Q p > 0
proof end;

theorem Th64: :: PREPOWER:64
for a being real number
for p, q being Rational st a > 0 holds
(a #Q p) * (a #Q q) = a #Q (p + q)
proof end;

theorem Th65: :: PREPOWER:65
for a being real number
for p being Rational st a > 0 holds
1 / (a #Q p) = a #Q (- p)
proof end;

theorem Th66: :: PREPOWER:66
for a being real number
for p, q being Rational st a > 0 holds
(a #Q p) / (a #Q q) = a #Q (p - q)
proof end;

theorem Th67: :: PREPOWER:67
for a, b being real number
for p being Rational st a > 0 & b > 0 holds
(a * b) #Q p = (a #Q p) * (b #Q p)
proof end;

theorem Th68: :: PREPOWER:68
for a being real number
for p being Rational st a > 0 holds
(1 / a) #Q p = 1 / (a #Q p)
proof end;

theorem Th69: :: PREPOWER:69
for a, b being real number
for p being Rational st a > 0 & b > 0 holds
(a / b) #Q p = (a #Q p) / (b #Q p)
proof end;

theorem Th70: :: PREPOWER:70
for a being real number
for p, q being Rational st a > 0 holds
(a #Q p) #Q q = a #Q (p * q)
proof end;

theorem Th71: :: PREPOWER:71
for a being real number
for p being Rational st a >= 1 & p >= 0 holds
a #Q p >= 1
proof end;

theorem Th72: :: PREPOWER:72
for a being real number
for p being Rational st a >= 1 & p <= 0 holds
a #Q p <= 1
proof end;

theorem Th73: :: PREPOWER:73
for a being real number
for p being Rational st a > 1 & p > 0 holds
a #Q p > 1
proof end;

theorem Th74: :: PREPOWER:74
for a being real number
for p, q being Rational st a >= 1 & p >= q holds
a #Q p >= a #Q q
proof end;

theorem Th75: :: PREPOWER:75
for a being real number
for p, q being Rational st a > 1 & p > q holds
a #Q p > a #Q q
proof end;

theorem Th76: :: PREPOWER:76
for a being real number
for p being Rational st a > 0 & a < 1 & p > 0 holds
a #Q p < 1
proof end;

theorem :: PREPOWER:77
for a being real number
for p being Rational st a > 0 & a <= 1 & p <= 0 holds
a #Q p >= 1
proof end;

definition
let IT be Real_Sequence;
redefine attr IT is rational-valued means :Def6: :: PREPOWER:def 6
 for n being Element of NAT holds IT . n is Rational;
compatibility
( IT is rational-valued iff for n being Element of NAT holds IT . n is Rational )
proof end;
end;

:: deftheorem Def6 defines rational-valued PREPOWER:def 6 :
for IT being Real_Sequence holds
( IT is rational-valued iff for n being Element of NAT holds IT . n is Rational );

registration
cluster Relation-like NAT -defined REAL -valued Function-like V11() V14( NAT ) V15( NAT , REAL ) complex-valued ext-real-valued real-valued rational-valued Element of K21(K22(NAT,REAL));
existence
ex b1 being Real_Sequence st b1 is rational-valued
proof end;
end;

definition
mode Rational_Sequence is rational-valued Real_Sequence;
end;

registration
let s be Rational_Sequence;
let n be Element of NAT ;
cluster s . n -> rational ;
coherence
s . n is rational ;
end;

theorem :: PREPOWER:78
canceled;

theorem Th79: :: PREPOWER:79
for a being real number ex s being Rational_Sequence st
( s is convergent & lim s = a & ( for n being Element of NAT holds s . n <= a ) )
proof end;

theorem Th80: :: PREPOWER:80
for a being real number ex s being Rational_Sequence st
( s is convergent & lim s = a & ( for n being Element of NAT holds s . n >= a ) )
proof end;

definition
let a be real number ;
let s be Rational_Sequence;
func a #Q s -> Real_Sequence means :Def7: :: PREPOWER:def 7
 for n being Element of NAT holds it . n = a #Q (s . n);
existence
ex b1 being Real_Sequence st
for n being Element of NAT holds b1 . n = a #Q (s . n)
proof end;
uniqueness
for b1, b2 being Real_Sequence st ( for n being Element of NAT holds b1 . n = a #Q (s . n) ) & ( for n being Element of NAT holds b2 . n = a #Q (s . n) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def7 defines #Q PREPOWER:def 7 :
for a being real number
for s being Rational_Sequence
for b3 being Real_Sequence holds
( b3 = a #Q s iff for n being Element of NAT holds b3 . n = a #Q (s . n) );

Lm6: for s being Rational_Sequence
for a being real number st s is convergent & a >= 1 holds
a #Q s is convergent
proof end;

theorem :: PREPOWER:81
canceled;

theorem Th82: :: PREPOWER:82
for a being real number
for s being Rational_Sequence st s is convergent & a > 0 holds
a #Q s is convergent
proof end;

Lm7: for s1, s2 being Rational_Sequence
for a being real number st s1 is convergent & s2 is convergent & lim s1 = lim s2 & a >= 1 holds
lim (a #Q s1) = lim (a #Q s2)
proof end;

theorem Th83: :: PREPOWER:83
for s1, s2 being Rational_Sequence
for a being real number st s1 is convergent & s2 is convergent & lim s1 = lim s2 & a > 0 holds
( a #Q s1 is convergent & a #Q s2 is convergent & lim (a #Q s1) = lim (a #Q s2) )
proof end;

definition
let a, b be real number ;
assume A1: a > 0 ;
func a #R b -> real number means :Def8: :: PREPOWER:def 8
 ex s being Rational_Sequence st
( s is convergent & lim s = b & a #Q s is convergent & lim (a #Q s) = it );
existence
ex b1 being real number ex s being Rational_Sequence st
( s is convergent & lim s = b & a #Q s is convergent & lim (a #Q s) = b1 )
proof end;
uniqueness
for b1, b2 being real number st ex s being Rational_Sequence st
( s is convergent & lim s = b & a #Q s is convergent & lim (a #Q s) = b1 ) & ex s being Rational_Sequence st
( s is convergent & lim s = b & a #Q s is convergent & lim (a #Q s) = b2 ) holds
b1 = b2 by A1, Th83;
end;

:: deftheorem Def8 defines #R PREPOWER:def 8 :
for a, b being real number st a > 0 holds
for b3 being real number holds
( b3 = a #R b iff ex s being Rational_Sequence st
( s is convergent & lim s = b & a #Q s is convergent & lim (a #Q s) = b3 ) );

definition
let a, b be Real;
:: original: #R
redefine func a #R b -> Real;
coherence
a #R b is Real by XREAL_0:def 1;
end;

theorem :: PREPOWER:84
canceled;

theorem Th85: :: PREPOWER:85
for a being real number st a > 0 holds
a #R 0 = 1
proof end;

theorem :: PREPOWER:86
for a being real number st a > 0 holds
a #R 1 = a
proof end;

theorem Th87: :: PREPOWER:87
for a being real number holds 1 #R a = 1
proof end;

theorem Th88: :: PREPOWER:88
for a being real number
for p being Rational st a > 0 holds
a #R p = a #Q p
proof end;

theorem Th89: :: PREPOWER:89
for a, b, c being real number st a > 0 holds
a #R (b + c) = (a #R b) * (a #R c)
proof end;

Lm8: for a, b being real number st a > 0 holds
a #R b >= 0
proof end;

Lm9: for a, b being real number st a > 0 holds
a #R b <> 0
proof end;

theorem Th90: :: PREPOWER:90
for a, c being real number st a > 0 holds
a #R (- c) = 1 / (a #R c)
proof end;

theorem Th91: :: PREPOWER:91
for a, b, c being real number st a > 0 holds
a #R (b - c) = (a #R b) / (a #R c)
proof end;

theorem Th92: :: PREPOWER:92
for a, b, c being real number st a > 0 & b > 0 holds
(a * b) #R c = (a #R c) * (b #R c)
proof end;

theorem Th93: :: PREPOWER:93
for a, c being real number st a > 0 holds
(1 / a) #R c = 1 / (a #R c)
proof end;

theorem :: PREPOWER:94
for a, b, c being real number st a > 0 & b > 0 holds
(a / b) #R c = (a #R c) / (b #R c)
proof end;

theorem Th95: :: PREPOWER:95
for a, b being real number st a > 0 holds
a #R b > 0
proof end;

theorem Th96: :: PREPOWER:96
for a, c, b being real number st a >= 1 & c >= b holds
a #R c >= a #R b
proof end;

theorem Th97: :: PREPOWER:97
for a, c, b being real number st a > 1 & c > b holds
a #R c > a #R b
proof end;

theorem Th98: :: PREPOWER:98
for a, c, b being real number st a > 0 & a <= 1 & c >= b holds
a #R c <= a #R b
proof end;

theorem Th99: :: PREPOWER:99
for a, b being real number st a >= 1 & b >= 0 holds
a #R b >= 1
proof end;

theorem Th100: :: PREPOWER:100
for a, b being real number st a > 1 & b > 0 holds
a #R b > 1
proof end;

theorem Th101: :: PREPOWER:101
for a, b being real number st a >= 1 & b <= 0 holds
a #R b <= 1
proof end;

theorem :: PREPOWER:102
for a, b being real number st a > 1 & b < 0 holds
a #R b < 1
proof end;

Lm10: for p being Rational
for s1, s2 being Real_Sequence st s1 is convergent & s2 is convergent & lim s1 > 0 & p >= 0 & ( for n being Element of NAT holds
( s1 . n > 0 & s2 . n = (s1 . n) #Q p ) ) holds
lim s2 = (lim s1) #Q p
proof end;

theorem Th103: :: PREPOWER:103
for p being Rational
for s1, s2 being Real_Sequence st s1 is convergent & s2 is convergent & lim s1 > 0 & ( for n being Element of NAT holds
( s1 . n > 0 & s2 . n = (s1 . n) #Q p ) ) holds
lim s2 = (lim s1) #Q p
proof end;

Lm11: for a, b being real number
for p being Rational st a > 0 holds
(a #R b) #Q p = a #R (b * p)
proof end;

Lm12: for a being real number
for s1, s2 being Real_Sequence st a >= 1 & s1 is convergent & s2 is convergent & ( for n being Element of NAT holds s2 . n = a #R (s1 . n) ) holds
lim s2 = a #R (lim s1)
proof end;

theorem Th104: :: PREPOWER:104
for a being real number
for s1, s2 being Real_Sequence st a > 0 & s1 is convergent & s2 is convergent & ( for n being Element of NAT holds s2 . n = a #R (s1 . n) ) holds
lim s2 = a #R (lim s1)
proof end;

theorem :: PREPOWER:105
for a, b, c being real number st a > 0 holds
(a #R b) #R c = a #R (b * c)
proof end;

begin

theorem :: PREPOWER:106
for r, u being real number st r > 0 & u > 0 holds
ex k being Element of NAT st u / (2 |^ k) <= r
proof end;

theorem :: PREPOWER:107
for n being Element of NAT
for r being real number
for k being natural number st k >= n & r >= 1 holds
r |^ k >= r |^ n
proof end;

theorem Th108: :: PREPOWER:108
for n, m, l being Element of NAT st n divides m & n divides l holds
n divides m - l
proof end;

theorem :: PREPOWER:109
for m, n being Element of NAT holds
( m divides n iff m divides n ) ;

theorem Th110: :: PREPOWER:110
for m, n being Element of NAT holds m gcd n = m gcd (abs (n - m))
proof end;

theorem :: PREPOWER:111
for a, b being Integer st a >= 0 & b >= 0 holds
a gcd b = a gcd (b - a)
proof end;

theorem :: PREPOWER:112
for a being real number
for l being Integer st a >= 0 holds
a #Z l >= 0 by Lm5;

theorem :: PREPOWER:113
for a being real number
for l being Integer st a > 0 holds
a #Q l = a #Z l
proof end;

theorem :: PREPOWER:114
for l being Integer st l <> 0 holds
0 #Z l = 0
proof end;