# Practise proofs

87. K(+,*,0,1,/), 1*a = a

86. K(+,*,0,1,/), (b+c)a = ba + ca

21. K(+,*,0,1,/), ∀ b a+b= b ⇒ a=0

22. K(+,*,0,1,/), ∃ b a+b = b ⇒ a=0

23. K(+,*,0,1,/), 0+0 =0

24. K(+,*,0,1,/), 0 = 0+0

25. K(+,*,0,1,/), a+a = a ⇒ a=0

27. K(+,*,0,1,/), -0 = 0

28. K(+,*,0,1,/), a + b = 0 ⇒ a = -b

36. K(+,*,0,1,/), a + b ≠ 0 ⇒ a ≠ -b

78. K(+,*,0,1,/), -(-a) = a

31. K(+,*,0,1,/), a = -b ⇒ a + b = 0

35. K(+,*,0,1,/), a = -b ⇒ b + a = 0

52. K(+,*,0,1,/), b+a = c+a ⇒ b=c

51. K(+,*,0,1,/), a+b = a+c ⇒ b=c

53. K(+,*,0,1,/), a+b = c+a ⇒ b=c

54. K(+,*,0,1,/), (0+0)+0 = 0

56. K(+,*,0,1,/), ∀ b ab = b ⇒ a=1

57. K(+,*,0,1,/), ∃ b ≠ 0, ab=b ⇒ a=1

58. K(+,*,0,1,/), 1*1 = 1

59. K(+,*,0,1,/), a*a=a, a≠ 0 ⇒ a=1

61. K(+,*,0,1,/), 1-1 = 1

82. K(+,*,0,1,/), a*0 = 0

85. K(+,*,0,1,/), 0*a = 0

69. K(+,*,0,1,/), ab = ac, a ≠ 0 ⇒ b=c

70. K(+,*,0,1,/), ba = ca, a ≠ 0 ⇒ b=c

71. K(+,*,0,1,/), ab = ca, a ≠ 0 ⇒ b=c

72. K(+,*,0,1,/), (-)(+) = (-)

73. K(+,*,0,1,/), (+)(-) = (-)

74. K(+,*,0,1,/), (-)(-)=(+)

75. K(+,*,0,1,/), (-1)a = -a

76. K(+,*,0,1,/), (-1)(-1) = 1

77. K(+,*,0,1,/), -(-1) = 1

79. K(+,*,0,1,/), -(-(-a)) = -a

43. f:VxV → K, f(cv+u,w) = cf(v,w) + f(u,w)

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