:: GCD_1 semantic presentation
theorem Th1: :: GCD_1:1
:: deftheorem Def1 defines divides GCD_1:def 1 :
:: deftheorem Def2 defines unital GCD_1:def 2 :
:: deftheorem Def3 defines is_associated_to GCD_1:def 3 :
:: deftheorem Def4 defines / GCD_1:def 4 :
theorem Th2: :: GCD_1:2
theorem Th3: :: GCD_1:3
theorem Th4: :: GCD_1:4
theorem Th5: :: GCD_1:5
theorem Th6: :: GCD_1:6
theorem Th7: :: GCD_1:7
theorem Th8: :: GCD_1:8
theorem Th9: :: GCD_1:9
theorem Th10: :: GCD_1:10
theorem Th11: :: GCD_1:11
theorem Th12: :: GCD_1:12
theorem Th13: :: GCD_1:13
theorem Th14: :: GCD_1:14
theorem Th15: :: GCD_1:15
theorem Th16: :: GCD_1:16
theorem Th17: :: GCD_1:17
theorem Th18: :: GCD_1:18
theorem Th19: :: GCD_1:19
:: deftheorem Def5 defines Class GCD_1:def 5 :
theorem Th20: :: GCD_1:20
:: deftheorem Def6 defines Classes GCD_1:def 6 :
theorem Th21: :: GCD_1:21
:: deftheorem Def7 defines Am GCD_1:def 7 :
:: deftheorem Def8 defines AmpleSet GCD_1:def 8 :
theorem Th22: :: GCD_1:22
theorem Th23: :: GCD_1:23
theorem Th24: :: GCD_1:24
:: deftheorem Def9 defines NF GCD_1:def 9 :
theorem Th25: :: GCD_1:25
theorem Th26: :: GCD_1:26
:: deftheorem Def10 defines multiplicative GCD_1:def 10 :
theorem Th27: :: GCD_1:27
:: deftheorem Def11 defines gcd-like GCD_1:def 11 :
:: deftheorem Def12 defines gcd GCD_1:def 12 :
theorem Th28: :: GCD_1:28
canceled;
theorem Th29: :: GCD_1:29
theorem Th30: :: GCD_1:30
theorem Th31: :: GCD_1:31
theorem Th32: :: GCD_1:32
theorem Th33: :: GCD_1:33
theorem Th34: :: GCD_1:34
theorem Th35: :: GCD_1:35
for b
1 being
gcdDomainfor b
2 being
AmpleSet of b
1for b
3, b
4, b
5 being
Element of b
1 holds
( b
4 is_associated_to b
5 implies (
gcd b
3,b
4,b
2 is_associated_to gcd b
3,b
5,b
2 &
gcd b
4,b
3,b
2 is_associated_to gcd b
5,b
3,b
2 ) )
theorem Th36: :: GCD_1:36
for b
1 being
gcdDomainfor b
2 being
AmpleSet of b
1for b
3, b
4, b
5 being
Element of b
1 holds
gcd (gcd b3,b4,b2),b
5,b
2 = gcd b
3,
(gcd b4,b5,b2),b
2
theorem Th37: :: GCD_1:37
theorem Th38: :: GCD_1:38
theorem Th39: :: GCD_1:39
theorem Th40: :: GCD_1:40
theorem Th41: :: GCD_1:41
for b
1 being
gcdDomainfor b
2 being
AmpleSet of b
1for b
3, b
4, b
5, b
6 being
Element of b
1 holds
(
gcd b
3,b
4,b
2 = 1. b
1 &
gcd b
5,b
6,b
2 = 1. b
1 & b
4 <> 0. b
1 & b
6 <> 0. b
1 implies
gcd ((b3 * (b6 / (gcd b4,b6,b2))) + (b5 * (b4 / (gcd b4,b6,b2)))),
(b4 * (b6 / (gcd b4,b6,b2))),b
2 = gcd ((b3 * (b6 / (gcd b4,b6,b2))) + (b5 * (b4 / (gcd b4,b6,b2)))),
(gcd b4,b6,b2),b
2 )
theorem Th42: :: GCD_1:42
for b
1 being
gcdDomainfor b
2 being
AmpleSet of b
1for b
3, b
4, b
5, b
6 being
Element of b
1 holds
(
gcd b
3,b
4,b
2 = 1. b
1 &
gcd b
5,b
6,b
2 = 1. b
1 & b
4 <> 0. b
1 & b
6 <> 0. b
1 implies
gcd ((b3 / (gcd b3,b6,b2)) * (b5 / (gcd b5,b4,b2))),
((b4 / (gcd b5,b4,b2)) * (b6 / (gcd b3,b6,b2))),b
2 = 1. b
1 )
:: deftheorem Def13 defines are_canonical_wrt GCD_1:def 13 :
theorem Th43: :: GCD_1:43
:: deftheorem Def14 defines are_co-prime GCD_1:def 14 :
theorem Th44: :: GCD_1:44
:: deftheorem Def15 defines are_normalized_wrt GCD_1:def 15 :
definition
let c
1 be
gcdDomain;
let c
2 be
AmpleSet of c
1;
let c
3, c
4, c
5, c
6 be
Element of c
1;
assume E56:
( c
3,c
4 are_co-prime & c
5,c
6 are_co-prime & c
4 = NF c
4,c
2 & c
6 = NF c
6,c
2 )
;
func add1 c
3,c
4,c
5,c
6,c
2 -> Element of a
1 equals :
Def16:
:: GCD_1:def 16
a
5 if a
3 = 0. a
1a
3 if a
5 = 0. a
1(a3 * a6) + (a4 * a5) if gcd a
4,a
6,a
2 = 1. a
1 0. a
1 if (a3 * (a6 / (gcd a4,a6,a2))) + (a5 * (a4 / (gcd a4,a6,a2))) = 0. a
1 otherwise ((a3 * (a6 / (gcd a4,a6,a2))) + (a5 * (a4 / (gcd a4,a6,a2)))) / (gcd ((a3 * (a6 / (gcd a4,a6,a2))) + (a5 * (a4 / (gcd a4,a6,a2)))),(gcd a4,a6,a2),a2);
coherence
( ( c3 = 0. c1 implies c5 is Element of c1 ) & ( c5 = 0. c1 implies c3 is Element of c1 ) & ( gcd c4,c6,c2 = 1. c1 implies (c3 * c6) + (c4 * c5) is Element of c1 ) & ( (c3 * (c6 / (gcd c4,c6,c2))) + (c5 * (c4 / (gcd c4,c6,c2))) = 0. c1 implies 0. c1 is Element of c1 ) & ( not c3 = 0. c1 & not c5 = 0. c1 & not gcd c4,c6,c2 = 1. c1 & not (c3 * (c6 / (gcd c4,c6,c2))) + (c5 * (c4 / (gcd c4,c6,c2))) = 0. c1 implies ((c3 * (c6 / (gcd c4,c6,c2))) + (c5 * (c4 / (gcd c4,c6,c2)))) / (gcd ((c3 * (c6 / (gcd c4,c6,c2))) + (c5 * (c4 / (gcd c4,c6,c2)))),(gcd c4,c6,c2),c2) is Element of c1 ) )
;
consistency
for b1 being Element of c1 holds
( ( c3 = 0. c1 & c5 = 0. c1 implies ( b1 = c5 iff b1 = c3 ) ) & ( c3 = 0. c1 & gcd c4,c6,c2 = 1. c1 implies ( b1 = c5 iff b1 = (c3 * c6) + (c4 * c5) ) ) & ( c3 = 0. c1 & (c3 * (c6 / (gcd c4,c6,c2))) + (c5 * (c4 / (gcd c4,c6,c2))) = 0. c1 implies ( b1 = c5 iff b1 = 0. c1 ) ) & ( c5 = 0. c1 & gcd c4,c6,c2 = 1. c1 implies ( b1 = c3 iff b1 = (c3 * c6) + (c4 * c5) ) ) & ( c5 = 0. c1 & (c3 * (c6 / (gcd c4,c6,c2))) + (c5 * (c4 / (gcd c4,c6,c2))) = 0. c1 implies ( b1 = c3 iff b1 = 0. c1 ) ) & ( gcd c4,c6,c2 = 1. c1 & (c3 * (c6 / (gcd c4,c6,c2))) + (c5 * (c4 / (gcd c4,c6,c2))) = 0. c1 implies ( b1 = (c3 * c6) + (c4 * c5) iff b1 = 0. c1 ) ) )
end;
:: deftheorem Def16 defines add1 GCD_1:def 16 :
for b
1 being
gcdDomainfor b
2 being
AmpleSet of b
1for b
3, b
4, b
5, b
6 being
Element of b
1 holds
( b
3,b
4 are_co-prime & b
5,b
6 are_co-prime & b
4 = NF b
4,b
2 & b
6 = NF b
6,b
2 implies ( ( b
3 = 0. b
1 implies
add1 b
3,b
4,b
5,b
6,b
2 = b
5 ) & ( b
5 = 0. b
1 implies
add1 b
3,b
4,b
5,b
6,b
2 = b
3 ) & (
gcd b
4,b
6,b
2 = 1. b
1 implies
add1 b
3,b
4,b
5,b
6,b
2 = (b3 * b6) + (b4 * b5) ) & (
(b3 * (b6 / (gcd b4,b6,b2))) + (b5 * (b4 / (gcd b4,b6,b2))) = 0. b
1 implies
add1 b
3,b
4,b
5,b
6,b
2 = 0. b
1 ) & ( not b
3 = 0. b
1 & not b
5 = 0. b
1 & not
gcd b
4,b
6,b
2 = 1. b
1 & not
(b3 * (b6 / (gcd b4,b6,b2))) + (b5 * (b4 / (gcd b4,b6,b2))) = 0. b
1 implies
add1 b
3,b
4,b
5,b
6,b
2 = ((b3 * (b6 / (gcd b4,b6,b2))) + (b5 * (b4 / (gcd b4,b6,b2)))) / (gcd ((b3 * (b6 / (gcd b4,b6,b2))) + (b5 * (b4 / (gcd b4,b6,b2)))),(gcd b4,b6,b2),b2) ) ) );
definition
let c
1 be
gcdDomain;
let c
2 be
AmpleSet of c
1;
let c
3, c
4, c
5, c
6 be
Element of c
1;
assume E57:
( c
3,c
4 are_co-prime & c
5,c
6 are_co-prime & c
4 = NF c
4,c
2 & c
6 = NF c
6,c
2 )
;
func add2 c
3,c
4,c
5,c
6,c
2 -> Element of a
1 equals :
Def17:
:: GCD_1:def 17
a
6 if a
3 = 0. a
1a
4 if a
5 = 0. a
1a
4 * a
6 if gcd a
4,a
6,a
2 = 1. a
1 1. a
1 if (a3 * (a6 / (gcd a4,a6,a2))) + (a5 * (a4 / (gcd a4,a6,a2))) = 0. a
1 otherwise (a4 * (a6 / (gcd a4,a6,a2))) / (gcd ((a3 * (a6 / (gcd a4,a6,a2))) + (a5 * (a4 / (gcd a4,a6,a2)))),(gcd a4,a6,a2),a2);
coherence
( ( c3 = 0. c1 implies c6 is Element of c1 ) & ( c5 = 0. c1 implies c4 is Element of c1 ) & ( gcd c4,c6,c2 = 1. c1 implies c4 * c6 is Element of c1 ) & ( (c3 * (c6 / (gcd c4,c6,c2))) + (c5 * (c4 / (gcd c4,c6,c2))) = 0. c1 implies 1. c1 is Element of c1 ) & ( not c3 = 0. c1 & not c5 = 0. c1 & not gcd c4,c6,c2 = 1. c1 & not (c3 * (c6 / (gcd c4,c6,c2))) + (c5 * (c4 / (gcd c4,c6,c2))) = 0. c1 implies (c4 * (c6 / (gcd c4,c6,c2))) / (gcd ((c3 * (c6 / (gcd c4,c6,c2))) + (c5 * (c4 / (gcd c4,c6,c2)))),(gcd c4,c6,c2),c2) is Element of c1 ) )
;
consistency
for b1 being Element of c1 holds
( ( c3 = 0. c1 & c5 = 0. c1 implies ( b1 = c6 iff b1 = c4 ) ) & ( c3 = 0. c1 & gcd c4,c6,c2 = 1. c1 implies ( b1 = c6 iff b1 = c4 * c6 ) ) & ( c3 = 0. c1 & (c3 * (c6 / (gcd c4,c6,c2))) + (c5 * (c4 / (gcd c4,c6,c2))) = 0. c1 implies ( b1 = c6 iff b1 = 1. c1 ) ) & ( c5 = 0. c1 & gcd c4,c6,c2 = 1. c1 implies ( b1 = c4 iff b1 = c4 * c6 ) ) & ( c5 = 0. c1 & (c3 * (c6 / (gcd c4,c6,c2))) + (c5 * (c4 / (gcd c4,c6,c2))) = 0. c1 implies ( b1 = c4 iff b1 = 1. c1 ) ) & ( gcd c4,c6,c2 = 1. c1 & (c3 * (c6 / (gcd c4,c6,c2))) + (c5 * (c4 / (gcd c4,c6,c2))) = 0. c1 implies ( b1 = c4 * c6 iff b1 = 1. c1 ) ) )
end;
:: deftheorem Def17 defines add2 GCD_1:def 17 :
for b
1 being
gcdDomainfor b
2 being
AmpleSet of b
1for b
3, b
4, b
5, b
6 being
Element of b
1 holds
( b
3,b
4 are_co-prime & b
5,b
6 are_co-prime & b
4 = NF b
4,b
2 & b
6 = NF b
6,b
2 implies ( ( b
3 = 0. b
1 implies
add2 b
3,b
4,b
5,b
6,b
2 = b
6 ) & ( b
5 = 0. b
1 implies
add2 b
3,b
4,b
5,b
6,b
2 = b
4 ) & (
gcd b
4,b
6,b
2 = 1. b
1 implies
add2 b
3,b
4,b
5,b
6,b
2 = b
4 * b
6 ) & (
(b3 * (b6 / (gcd b4,b6,b2))) + (b5 * (b4 / (gcd b4,b6,b2))) = 0. b
1 implies
add2 b
3,b
4,b
5,b
6,b
2 = 1. b
1 ) & ( not b
3 = 0. b
1 & not b
5 = 0. b
1 & not
gcd b
4,b
6,b
2 = 1. b
1 & not
(b3 * (b6 / (gcd b4,b6,b2))) + (b5 * (b4 / (gcd b4,b6,b2))) = 0. b
1 implies
add2 b
3,b
4,b
5,b
6,b
2 = (b4 * (b6 / (gcd b4,b6,b2))) / (gcd ((b3 * (b6 / (gcd b4,b6,b2))) + (b5 * (b4 / (gcd b4,b6,b2)))),(gcd b4,b6,b2),b2) ) ) );
theorem Th45: :: GCD_1:45
for b
1 being
gcdDomainfor b
2 being
AmpleSet of b
1for b
3, b
4, b
5, b
6 being
Element of b
1 holds
( b
2 is
multiplicative & b
3,b
4 are_normalized_wrt b
2 & b
5,b
6 are_normalized_wrt b
2 implies
add1 b
3,b
4,b
5,b
6,b
2,
add2 b
3,b
4,b
5,b
6,b
2 are_normalized_wrt b
2 )
theorem Th46: :: GCD_1:46
for b
1 being
gcdDomainfor b
2 being
AmpleSet of b
1for b
3, b
4, b
5, b
6 being
Element of b
1 holds
( b
2 is
multiplicative & b
3,b
4 are_normalized_wrt b
2 & b
5,b
6 are_normalized_wrt b
2 implies
(add1 b3,b4,b5,b6,b2) * (b4 * b6) = (add2 b3,b4,b5,b6,b2) * ((b3 * b6) + (b5 * b4)) )
definition
let c
1 be
gcdDomain;
let c
2 be
AmpleSet of c
1;
let c
3, c
4, c
5, c
6 be
Element of c
1;
func mult1 c
3,c
4,c
5,c
6,c
2 -> Element of a
1 equals :
Def18:
:: GCD_1:def 18
0. a
1 if ( a
3 = 0. a
1 or a
5 = 0. a
1 )
a
3 * a
5 if ( a
4 = 1. a
1 & a
6 = 1. a
1 )
(a3 * a5) / (gcd a3,a6,a2) if ( a
6 <> 0. a
1 & a
4 = 1. a
1 )
(a3 * a5) / (gcd a5,a4,a2) if ( a
4 <> 0. a
1 & a
6 = 1. a
1 )
otherwise (a3 / (gcd a3,a6,a2)) * (a5 / (gcd a5,a4,a2));
coherence
( ( ( c3 = 0. c1 or c5 = 0. c1 ) implies 0. c1 is Element of c1 ) & ( c4 = 1. c1 & c6 = 1. c1 implies c3 * c5 is Element of c1 ) & ( c6 <> 0. c1 & c4 = 1. c1 implies (c3 * c5) / (gcd c3,c6,c2) is Element of c1 ) & ( c4 <> 0. c1 & c6 = 1. c1 implies (c3 * c5) / (gcd c5,c4,c2) is Element of c1 ) & not ( not c3 = 0. c1 & not c5 = 0. c1 & not ( c4 = 1. c1 & c6 = 1. c1 ) & not ( c6 <> 0. c1 & c4 = 1. c1 ) & not ( c4 <> 0. c1 & c6 = 1. c1 ) & not (c3 / (gcd c3,c6,c2)) * (c5 / (gcd c5,c4,c2)) is Element of c1 ) )
;
consistency
for b1 being Element of c1 holds
( ( ( c3 = 0. c1 or c5 = 0. c1 ) & c4 = 1. c1 & c6 = 1. c1 implies ( b1 = 0. c1 iff b1 = c3 * c5 ) ) & ( ( c3 = 0. c1 or c5 = 0. c1 ) & c6 <> 0. c1 & c4 = 1. c1 implies ( b1 = 0. c1 iff b1 = (c3 * c5) / (gcd c3,c6,c2) ) ) & ( ( c3 = 0. c1 or c5 = 0. c1 ) & c4 <> 0. c1 & c6 = 1. c1 implies ( b1 = 0. c1 iff b1 = (c3 * c5) / (gcd c5,c4,c2) ) ) & ( c4 = 1. c1 & c6 = 1. c1 & c6 <> 0. c1 & c4 = 1. c1 implies ( b1 = c3 * c5 iff b1 = (c3 * c5) / (gcd c3,c6,c2) ) ) & ( c4 = 1. c1 & c6 = 1. c1 & c4 <> 0. c1 & c6 = 1. c1 implies ( b1 = c3 * c5 iff b1 = (c3 * c5) / (gcd c5,c4,c2) ) ) & ( c6 <> 0. c1 & c4 = 1. c1 & c4 <> 0. c1 & c6 = 1. c1 implies ( b1 = (c3 * c5) / (gcd c3,c6,c2) iff b1 = (c3 * c5) / (gcd c5,c4,c2) ) ) )
end;
:: deftheorem Def18 defines mult1 GCD_1:def 18 :
for b
1 being
gcdDomainfor b
2 being
AmpleSet of b
1for b
3, b
4, b
5, b
6 being
Element of b
1 holds
( ( ( b
3 = 0. b
1 or b
5 = 0. b
1 ) implies
mult1 b
3,b
4,b
5,b
6,b
2 = 0. b
1 ) & ( b
4 = 1. b
1 & b
6 = 1. b
1 implies
mult1 b
3,b
4,b
5,b
6,b
2 = b
3 * b
5 ) & ( b
6 <> 0. b
1 & b
4 = 1. b
1 implies
mult1 b
3,b
4,b
5,b
6,b
2 = (b3 * b5) / (gcd b3,b6,b2) ) & ( b
4 <> 0. b
1 & b
6 = 1. b
1 implies
mult1 b
3,b
4,b
5,b
6,b
2 = (b3 * b5) / (gcd b5,b4,b2) ) & not ( not b
3 = 0. b
1 & not b
5 = 0. b
1 & not ( b
4 = 1. b
1 & b
6 = 1. b
1 ) & not ( b
6 <> 0. b
1 & b
4 = 1. b
1 ) & not ( b
4 <> 0. b
1 & b
6 = 1. b
1 ) & not
mult1 b
3,b
4,b
5,b
6,b
2 = (b3 / (gcd b3,b6,b2)) * (b5 / (gcd b5,b4,b2)) ) );
definition
let c
1 be
gcdDomain;
let c
2 be
AmpleSet of c
1;
let c
3, c
4, c
5, c
6 be
Element of c
1;
assume E59:
( c
3,c
4 are_co-prime & c
5,c
6 are_co-prime & c
4 = NF c
4,c
2 & c
6 = NF c
6,c
2 )
;
func mult2 c
3,c
4,c
5,c
6,c
2 -> Element of a
1 equals :
Def19:
:: GCD_1:def 19
1. a
1 if ( a
3 = 0. a
1 or a
5 = 0. a
1 )
1. a
1 if ( a
4 = 1. a
1 & a
6 = 1. a
1 )
a
6 / (gcd a3,a6,a2) if ( a
6 <> 0. a
1 & a
4 = 1. a
1 )
a
4 / (gcd a5,a4,a2) if ( a
4 <> 0. a
1 & a
6 = 1. a
1 )
otherwise (a4 / (gcd a5,a4,a2)) * (a6 / (gcd a3,a6,a2));
coherence
( ( ( c3 = 0. c1 or c5 = 0. c1 ) implies 1. c1 is Element of c1 ) & ( c4 = 1. c1 & c6 = 1. c1 implies 1. c1 is Element of c1 ) & ( c6 <> 0. c1 & c4 = 1. c1 implies c6 / (gcd c3,c6,c2) is Element of c1 ) & ( c4 <> 0. c1 & c6 = 1. c1 implies c4 / (gcd c5,c4,c2) is Element of c1 ) & not ( not c3 = 0. c1 & not c5 = 0. c1 & not ( c4 = 1. c1 & c6 = 1. c1 ) & not ( c6 <> 0. c1 & c4 = 1. c1 ) & not ( c4 <> 0. c1 & c6 = 1. c1 ) & not (c4 / (gcd c5,c4,c2)) * (c6 / (gcd c3,c6,c2)) is Element of c1 ) )
;
consistency
for b1 being Element of c1 holds
( ( ( c3 = 0. c1 or c5 = 0. c1 ) & c4 = 1. c1 & c6 = 1. c1 implies ( b1 = 1. c1 iff b1 = 1. c1 ) ) & ( ( c3 = 0. c1 or c5 = 0. c1 ) & c6 <> 0. c1 & c4 = 1. c1 implies ( b1 = 1. c1 iff b1 = c6 / (gcd c3,c6,c2) ) ) & ( ( c3 = 0. c1 or c5 = 0. c1 ) & c4 <> 0. c1 & c6 = 1. c1 implies ( b1 = 1. c1 iff b1 = c4 / (gcd c5,c4,c2) ) ) & ( c4 = 1. c1 & c6 = 1. c1 & c6 <> 0. c1 & c4 = 1. c1 implies ( b1 = 1. c1 iff b1 = c6 / (gcd c3,c6,c2) ) ) & ( c4 = 1. c1 & c6 = 1. c1 & c4 <> 0. c1 & c6 = 1. c1 implies ( b1 = 1. c1 iff b1 = c4 / (gcd c5,c4,c2) ) ) & ( c6 <> 0. c1 & c4 = 1. c1 & c4 <> 0. c1 & c6 = 1. c1 implies ( b1 = c6 / (gcd c3,c6,c2) iff b1 = c4 / (gcd c5,c4,c2) ) ) )
end;
:: deftheorem Def19 defines mult2 GCD_1:def 19 :
for b
1 being
gcdDomainfor b
2 being
AmpleSet of b
1for b
3, b
4, b
5, b
6 being
Element of b
1 holds
( b
3,b
4 are_co-prime & b
5,b
6 are_co-prime & b
4 = NF b
4,b
2 & b
6 = NF b
6,b
2 implies ( ( ( b
3 = 0. b
1 or b
5 = 0. b
1 ) implies
mult2 b
3,b
4,b
5,b
6,b
2 = 1. b
1 ) & ( b
4 = 1. b
1 & b
6 = 1. b
1 implies
mult2 b
3,b
4,b
5,b
6,b
2 = 1. b
1 ) & ( b
6 <> 0. b
1 & b
4 = 1. b
1 implies
mult2 b
3,b
4,b
5,b
6,b
2 = b
6 / (gcd b3,b6,b2) ) & ( b
4 <> 0. b
1 & b
6 = 1. b
1 implies
mult2 b
3,b
4,b
5,b
6,b
2 = b
4 / (gcd b5,b4,b2) ) & not ( not b
3 = 0. b
1 & not b
5 = 0. b
1 & not ( b
4 = 1. b
1 & b
6 = 1. b
1 ) & not ( b
6 <> 0. b
1 & b
4 = 1. b
1 ) & not ( b
4 <> 0. b
1 & b
6 = 1. b
1 ) & not
mult2 b
3,b
4,b
5,b
6,b
2 = (b4 / (gcd b5,b4,b2)) * (b6 / (gcd b3,b6,b2)) ) ) );
theorem Th47: :: GCD_1:47
for b
1 being
gcdDomainfor b
2 being
AmpleSet of b
1for b
3, b
4, b
5, b
6 being
Element of b
1 holds
( b
2 is
multiplicative & b
3,b
4 are_normalized_wrt b
2 & b
5,b
6 are_normalized_wrt b
2 implies
mult1 b
3,b
4,b
5,b
6,b
2,
mult2 b
3,b
4,b
5,b
6,b
2 are_normalized_wrt b
2 )
theorem Th48: :: GCD_1:48
for b
1 being
gcdDomainfor b
2 being
AmpleSet of b
1for b
3, b
4, b
5, b
6 being
Element of b
1 holds
( b
2 is
multiplicative & b
3,b
4 are_normalized_wrt b
2 & b
5,b
6 are_normalized_wrt b
2 implies
(mult1 b3,b4,b5,b6,b2) * (b4 * b6) = (mult2 b3,b4,b5,b6,b2) * (b3 * b5) )
theorem Th49: :: GCD_1:49
canceled;
theorem Th50: :: GCD_1:50
canceled;
theorem Th51: :: GCD_1:51
theorem Th52: :: GCD_1:52
canceled;
theorem Th53: :: GCD_1:53