:: EULER_2 semantic presentation
Lemma33:
for a, b being Element of NAT holds a gcd b = a hcf b
Lemma34:
for t being Integer holds
( t < 1 iff t <= 0 )
Lemma37:
for a being Element of NAT st a <> 0 holds
a - 1 >= 0
Lemma38:
for z being Integer holds 1 gcd z = 1
theorem Th1: :: EULER_2:1
theorem Th2: :: EULER_2:2
Lemma41:
for m being Element of NAT
for z being Integer st m > 1 & 1 - m <= z & z <= m - 1 & m divides z holds
z = 0
Lemma48:
for m being Element of NAT
for t being Integer st m > 1 & m * t >= 0 holds
t >= 0
by REAL_2:145;
theorem Th3: :: EULER_2:3
canceled;
theorem Th4: :: EULER_2:4
canceled;
theorem Th5: :: EULER_2:5
theorem Th6: :: EULER_2:6
theorem Th7: :: EULER_2:7
theorem Th8: :: EULER_2:8
theorem Th9: :: EULER_2:9
theorem Th10: :: EULER_2:10
theorem Th11: :: EULER_2:11
Lemma71:
for f being FinSequence of NAT
for r being Element of NAT holds Product (f ^ <*r*>) = (Product f) * r
by RVSUM_1:126;
Lemma72:
for f1, f2 being FinSequence of NAT holds Product (f1 ^ f2) = (Product f1) * (Product f2)
by RVSUM_1:127;
theorem Th12: :: EULER_2:12
canceled;
theorem Th13: :: EULER_2:13
canceled;
theorem Th14: :: EULER_2:14
canceled;
theorem Th15: :: EULER_2:15
canceled;
theorem Th16: :: EULER_2:16
canceled;
theorem Th17: :: EULER_2:17
canceled;
theorem Th18: :: EULER_2:18
canceled;
theorem Th19: :: EULER_2:19
canceled;
theorem Th20: :: EULER_2:20
canceled;
theorem Th21: :: EULER_2:21
canceled;
theorem Th22: :: EULER_2:22
canceled;
theorem Th23: :: EULER_2:23
canceled;
theorem Th24: :: EULER_2:24
canceled;
theorem Th25: :: EULER_2:25
:: deftheorem Def1 defines mod EULER_2:def 1 :
theorem Th26: :: EULER_2:26
theorem Th27: :: EULER_2:27
theorem Th28: :: EULER_2:28
theorem Th29: :: EULER_2:29
theorem Th30: :: EULER_2:30
theorem Th31: :: EULER_2:31
theorem Th32: :: EULER_2:32
theorem Th33: :: EULER_2:33
theorem Th34: :: EULER_2:34