:: QUIN_1 semantic presentation
:: deftheorem Def1 defines delta QUIN_1:def 1 :
theorem Th1: :: QUIN_1:1
theorem Th2: :: QUIN_1:2
theorem Th3: :: QUIN_1:3
for b
1, b
2, b
3, b
4 being
real number holds
not ( b
1 > 0 &
delta b
1,b
2,b
3 < 0 & not
((b1 * (b4 ^2 )) + (b2 * b4)) + b
3 > 0 )
theorem Th4: :: QUIN_1:4
theorem Th5: :: QUIN_1:5
for b
1, b
2, b
3, b
4 being
real number holds
not ( b
1 < 0 &
delta b
1,b
2,b
3 < 0 & not
((b1 * (b4 ^2 )) + (b2 * b4)) + b
3 < 0 )
theorem Th6: :: QUIN_1:6
theorem Th7: :: QUIN_1:7
for b
1, b
2, b
3, b
4 being
real number holds
not ( b
1 > 0 &
((b1 * (b2 ^2 )) + (b3 * b2)) + b
4 > 0 & not
((((2 * b1) * b2) + b3) ^2 ) - (delta b1,b3,b4) > 0 )
theorem Th8: :: QUIN_1:8
theorem Th9: :: QUIN_1:9
for b
1, b
2, b
3, b
4 being
real number holds
not ( b
1 < 0 &
((b1 * (b2 ^2 )) + (b3 * b2)) + b
4 < 0 & not
((((2 * b1) * b2) + b3) ^2 ) - (delta b1,b3,b4) > 0 )
theorem Th10: :: QUIN_1:10
theorem Th11: :: QUIN_1:11
theorem Th12: :: QUIN_1:12
theorem Th13: :: QUIN_1:13
theorem Th14: :: QUIN_1:14
Lemma7:
for b1, b2 being complex number holds
not ( b1 ^2 = b2 ^2 & not b1 = b2 & not b1 = - b2 )
theorem Th15: :: QUIN_1:15
theorem Th16: :: QUIN_1:16
theorem Th17: :: QUIN_1:17
theorem Th18: :: QUIN_1:18
theorem Th19: :: QUIN_1:19
for b
1, b
2, b
3, b
4 being
real number holds
( b
1 < 0 &
delta b
1,b
2,b
3 > 0 implies ( not (
((b1 * (b4 ^2 )) + (b2 * b4)) + b
3 < 0 & not b
4 < ((- b2) + (sqrt (delta b1,b2,b3))) / (2 * b1) & not b
4 > ((- b2) - (sqrt (delta b1,b2,b3))) / (2 * b1) ) & not ( not ( not b
4 < ((- b2) + (sqrt (delta b1,b2,b3))) / (2 * b1) & not b
4 > ((- b2) - (sqrt (delta b1,b2,b3))) / (2 * b1) ) & not
((b1 * (b4 ^2 )) + (b2 * b4)) + b
3 < 0 ) ) )
theorem Th20: :: QUIN_1:20
canceled;
theorem Th21: :: QUIN_1:21
canceled;
theorem Th22: :: QUIN_1:22
theorem Th23: :: QUIN_1:23
for b
1, b
2, b
3, b
4 being
real number holds
not ( b
1 > 0 &
((((2 * b1) * b2) + b3) ^2 ) - (delta b1,b3,b4) > 0 & not
((b1 * (b2 ^2 )) + (b3 * b2)) + b
4 > 0 )
theorem Th24: :: QUIN_1:24
for b
1, b
2, b
3, b
4 being
real number holds
( b
1 > 0 &
delta b
1,b
2,b
3 = 0 implies ( not (
((b1 * (b4 ^2 )) + (b2 * b4)) + b
3 > 0 & not b
4 <> - (b2 / (2 * b1)) ) & not ( b
4 <> - (b2 / (2 * b1)) & not
((b1 * (b4 ^2 )) + (b2 * b4)) + b
3 > 0 ) ) )
theorem Th25: :: QUIN_1:25
for b
1, b
2, b
3, b
4 being
real number holds
not ( b
1 < 0 &
((((2 * b1) * b2) + b3) ^2 ) - (delta b1,b3,b4) > 0 & not
((b1 * (b2 ^2 )) + (b3 * b2)) + b
4 < 0 )
theorem Th26: :: QUIN_1:26
for b
1, b
2, b
3, b
4 being
real number holds
( b
1 < 0 &
delta b
1,b
2,b
3 = 0 implies ( not (
((b1 * (b4 ^2 )) + (b2 * b4)) + b
3 < 0 & not b
4 <> - (b2 / (2 * b1)) ) & not ( b
4 <> - (b2 / (2 * b1)) & not
((b1 * (b4 ^2 )) + (b2 * b4)) + b
3 < 0 ) ) )
theorem Th27: :: QUIN_1:27
theorem Th28: :: QUIN_1:28
theorem Th29: :: QUIN_1:29
for b
1, b
2, b
3, b
4 being
real number holds
( b
1 > 0 &
delta b
1,b
2,b
3 > 0 implies ( not (
((b1 * (b4 ^2 )) + (b2 * b4)) + b
3 > 0 & not b
4 < ((- b2) - (sqrt (delta b1,b2,b3))) / (2 * b1) & not b
4 > ((- b2) + (sqrt (delta b1,b2,b3))) / (2 * b1) ) & not ( not ( not b
4 < ((- b2) - (sqrt (delta b1,b2,b3))) / (2 * b1) & not b
4 > ((- b2) + (sqrt (delta b1,b2,b3))) / (2 * b1) ) & not
((b1 * (b4 ^2 )) + (b2 * b4)) + b
3 > 0 ) ) )