a + (b + 1) = (a + b) + 1     (R)


a + (b + (c + 1))
R
=
a + ((b + c)) + 1)
R
=
(a + (b + c)) + 1

(a + b) + (c + 1)                               
R
=
((a + b) + c) + 1
    

} a + (b + c) = (a + b) + c           (I)

    




(a + 1) + 1
i
=
(a + 1) + 1

1 + (a + 1)
R
=
(1 + a) + 1
         

} a + 1 = 1 + a …          (II)

    




a + (b + 1)
R
=
(a + b) + 1

(b + 1) + a
R
=
b + (1 + a)
II
=
b + (a + 1)
R
=
(b + a) + 1
         

} a + b = b + a           (III)

    
92



a ∙ 1 = a            (D)
a ∙ (b + 1) = a ∙ b + a           (M)

a ∙ (b + (c + 1))
R
=
a ∙ ((b + c) + 1)
M
=
a ∙ (b + c) + a

a ∙ b + (a ∙ (c + 1))
M
=
a ∙ b + (a ∙ c + a)
I
=
(a ∙ b + a ∙ c) + a
    
} a ∙ (b + c) = a ∙ b + a ∙ c           (IV)