:: PYTHTRIP semantic presentation
:: deftheorem Def1 defines are_relative_prime PYTHTRIP:def 1 :
:: deftheorem Def2 defines are_relative_prime PYTHTRIP:def 2 :
:: deftheorem Def3 defines square PYTHTRIP:def 3 :
theorem Th1: :: PYTHTRIP:1
theorem Th2: :: PYTHTRIP:2
theorem Th3: :: PYTHTRIP:3
theorem Th4: :: PYTHTRIP:4
theorem Th5: :: PYTHTRIP:5
for b
1, b
2 being
Nat holds
( b
1 ^2 = b
2 ^2 implies b
1 = b
2 )
theorem Th6: :: PYTHTRIP:6
theorem Th7: :: PYTHTRIP:7
theorem Th8: :: PYTHTRIP:8
for b
1, b
2, b
3 being
Nat holds
(b1 * b2) hcf (b1 * b3) = b
1 * (b2 hcf b3)
theorem Th9: :: PYTHTRIP:9
for b
1 being
set holds
not ( ( for b
2 being
Nat holds
ex b
3 being
Nat st
( b
3 >= b
2 & b
3 in b
1 ) ) & b
1 is
finite )
theorem Th10: :: PYTHTRIP:10
theorem Th11: :: PYTHTRIP:11
theorem Th12: :: PYTHTRIP:12
for b
1, b
2, b
3, b
4, b
5 being
Nat holds
( b
1 = (b2 ^2 ) - (b3 ^2 ) & b
4 = (2 * b3) * b
2 & b
5 = (b2 ^2 ) + (b3 ^2 ) implies
(b1 ^2 ) + (b4 ^2 ) = b
5 ^2 ) ;
:: deftheorem Def4 defines Pythagorean_triple PYTHTRIP:def 4 :
:: deftheorem Def5 defines Pythagorean_triple PYTHTRIP:def 5 :
:: deftheorem Def6 defines degenerate PYTHTRIP:def 6 :
theorem Th13: :: PYTHTRIP:13
:: deftheorem Def7 defines simplified PYTHTRIP:def 7 :
:: deftheorem Def8 defines simplified PYTHTRIP:def 8 :
theorem Th14: :: PYTHTRIP:14
theorem Th15: :: PYTHTRIP:15
theorem Th16: :: PYTHTRIP:16