:: SERIES_5 semantic presentation
Lemma31:
for x, y being real number holds (x |^ 3) - (y |^ 3) = (x - y) * (((x ^2 ) + (x * y)) + (y ^2 ))
Lemma32:
for a, b being positive real number holds 2 / ((1 / a) + (1 / b)) = (2 * (a * b)) / (a + b)
Lemma33:
for x, y, z being real number holds ((1 / x) * (1 / y)) * (1 / z) = 1 / ((x * y) * z)
Lemma34:
for a, c being positive real number
for b being real number holds (a / c) to_power (- b) = (c / a) to_power b
Lemma35:
for a, b, c, d being positive real number holds (((sqrt (a * b)) ^2 ) + ((sqrt (c * d)) ^2 )) * (((sqrt (a * c)) ^2 ) + ((sqrt (b * d)) ^2 )) >= (b * c) * ((a + d) ^2 )
Lemma38:
for x, y, z being real number holds ((x + y) + z) ^2 = (((((x ^2 ) + (y ^2 )) + (z ^2 )) + ((2 * x) * y)) + ((2 * y) * z)) + ((2 * z) * x)
;
Lemma39:
for x being real number st abs x < 1 holds
x ^2 < 1
Lemma40:
for x being real number st x ^2 < 1 holds
abs x < 1
Lemma41:
for a, b, c being positive real number holds (((((((2 * (a ^2 )) * (sqrt (b * c))) * 2) * (b ^2 )) * (sqrt (a * c))) * 2) * (c ^2 )) * (sqrt (a * b)) = (((2 * a) * b) * c) |^ 3
Lemma42:
for a, b being positive real number holds sqrt (((a ^2 ) + (a * b)) + (b ^2 )) = (1 / 2) * (sqrt ((4 * ((a ^2 ) + (b ^2 ))) + ((4 * a) * b)))
Lemma46:
for a, b being positive real number holds sqrt (((a ^2 ) + (a * b)) + (b ^2 )) >= ((1 / 2) * (sqrt 3)) * (a + b)
Lemma47:
for a, b being positive real number holds sqrt ((((a ^2 ) + (a * b)) + (b ^2 )) / 3) <= sqrt (((a ^2 ) + (b ^2 )) / 2)
Lemma48:
for b, c, a being positive real number holds (((b * c) / a) ^2 ) + (((c * a) / b) ^2 ) >= 2 * (c ^2 )
Lemma49:
for b, c, a being positive real number holds ((b * c) / a) * ((c * a) / b) = c ^2
Lemma50:
for b, c, a being positive real number holds (((2 * ((b * c) / a)) * ((c * a) / b)) + ((2 * ((b * c) / a)) * ((a * b) / c))) + ((2 * ((c * a) / b)) * ((a * b) / c)) = 2 * (((a ^2 ) + (b ^2 )) + (c ^2 ))
Lemma51:
for b, c, a being positive real number holds ((((b * c) / a) + ((c * a) / b)) + ((a * b) / c)) ^2 = (((((c * a) / b) ^2 ) + (((a * b) / c) ^2 )) + (((b * c) / a) ^2 )) + (2 * (((a ^2 ) + (b ^2 )) + (c ^2 )))
Lemma52:
for a, b, c being positive real number st (a + b) + c = 1 holds
(1 / a) - 1 = (b + c) / a
Lemma53:
for b, c, a being positive real number holds (((2 * (sqrt (b * c))) / a) * ((2 * (sqrt (a * c))) / b)) * ((2 * (sqrt (a * b))) / c) = 8
Lemma54:
for a, b, c being positive real number st (a + b) + c = 1 holds
1 + (1 / a) = 2 + ((b + c) / a)
Lemma55:
for a, c, b being positive real number holds (1 + ((sqrt (a * c)) / b)) * ((sqrt (a * b)) / c) = ((sqrt (a * b)) / c) + (a / (sqrt (b * c)))
Lemma56:
for b, c, a being positive real number holds (((sqrt (b * c)) / a) + (c / (sqrt (b * a)))) * ((sqrt (a * b)) / c) = (b / (sqrt (a * c))) + 1
Lemma57:
for b, c, a being positive real number holds ((1 + ((sqrt (b * c)) / a)) * (1 + ((sqrt (a * c)) / b))) * (1 + ((sqrt (a * b)) / c)) = (((((2 + ((sqrt (a * c)) / b)) + (b / (sqrt (a * c)))) + ((sqrt (a * b)) / c)) + (c / (sqrt (b * a)))) + (a / (sqrt (b * c)))) + ((sqrt (b * c)) / a)
Lemma58:
for a, c, b being positive real number holds ((((((sqrt (a * c)) / b) + (b / (sqrt (a * c)))) + ((sqrt (a * b)) / c)) + (c / (sqrt (b * a)))) + (a / (sqrt (b * c)))) + ((sqrt (b * c)) / a) >= 6
theorem Th1: :: SERIES_5:1
theorem Th2: :: SERIES_5:2
theorem Th3: :: SERIES_5:3
theorem Th4: :: SERIES_5:4
theorem Th5: :: SERIES_5:5
theorem Th6: :: SERIES_5:6
theorem Th7: :: SERIES_5:7
theorem Th8: :: SERIES_5:8
theorem Th9: :: SERIES_5:9
theorem Th10: :: SERIES_5:10
theorem Th11: :: SERIES_5:11
theorem Th12: :: SERIES_5:12
theorem Th13: :: SERIES_5:13
theorem Th14: :: SERIES_5:14
theorem Th15: :: SERIES_5:15
theorem Th16: :: SERIES_5:16
theorem Th17: :: SERIES_5:17
theorem Th18: :: SERIES_5:18
theorem Th19: :: SERIES_5:19
theorem Th20: :: SERIES_5:20
theorem Th21: :: SERIES_5:21
theorem Th22: :: SERIES_5:22
theorem Th23: :: SERIES_5:23
theorem Th24: :: SERIES_5:24
theorem Th25: :: SERIES_5:25
theorem Th26: :: SERIES_5:26
theorem Th27: :: SERIES_5:27
theorem Th28: :: SERIES_5:28
theorem Th29: :: SERIES_5:29
theorem Th30: :: SERIES_5:30
theorem Th31: :: SERIES_5:31
theorem Th32: :: SERIES_5:32
theorem Th33: :: SERIES_5:33
theorem Th34: :: SERIES_5:34
theorem Th35: :: SERIES_5:35
theorem Th36: :: SERIES_5:36
theorem Th37: :: SERIES_5:37
theorem Th38: :: SERIES_5:38
theorem Th39: :: SERIES_5:39
theorem Th40: :: SERIES_5:40
theorem Th41: :: SERIES_5:41
theorem Th42: :: SERIES_5:42
theorem Th43: :: SERIES_5:43
theorem Th44: :: SERIES_5:44
theorem Th45: :: SERIES_5:45
theorem Th46: :: SERIES_5:46
theorem Th47: :: SERIES_5:47
theorem Th48: :: SERIES_5:48
theorem Th49: :: SERIES_5:49
theorem Th50: :: SERIES_5:50
theorem Th51: :: SERIES_5:51
theorem Th52: :: SERIES_5:52
theorem Th53: :: SERIES_5:53
theorem Th54: :: SERIES_5:54
theorem Th55: :: SERIES_5:55
theorem Th56: :: SERIES_5:56
theorem Th57: :: SERIES_5:57