a+b=b

a*(a+b)=a*b

A*(a+b)=a (absorption rate)

a*b=a

And a*b≤b

a≤b

Conversely, from a≤b, b ≤ b.

? a+b≤b

And a+b≥b

A+b=b (antisymmetry)

therefore

a≤b？ a+b=b

2

(1) from a≤b? a+b=b

And b≤c? B*c=b (similar to the proof of 1

a+b = b*c

(2)

a*b=a

b*c=b

a+b=b

a+c=c

rule

(a*b)+(b*c)=a+b=b

(a+b)*(b+c)=b*c=b

therefore

(a*b)+(b*c)=b=(a+b)*(b+c)

three

a*c≤a≤b

a*c≤c≤d

Through transitivity, we get

a*c≤b

a*c≤d

Then a*c≤b*d

four

Obviously, in the penultimate line, the outermost two elements A and B have A and B.

But in

therefore