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proof with The following features: Trichotomy, Transitivity, Connection compatibility, Compatibility with multiplication 1.∀x, y ∈ Q...

proof with The following features: Trichotomy, Transitivity, Connection compatibility, Compatibility with multiplication

1.∀x, y ∈ Q xy>0 ⇐⇒ (y > 0 and x > 0) or ( 0 > y and 0 > x)

2.∀ x ∈ Q x^2 > 0 ⇐⇒ x is not equal to 0

3. ∀ x, y, z, w∈ Q (x < y) ∧(z < w) ⇒ x+z < y+w

4.∀x, y, z, w∈ Q   (0 < x < y) ∧(0 < z < w) ⇒ xz < yw

5. ∀ x,y ∈ Q

6. ∀x,y∈ Q 0 < x < y ⇒ 0 < 1/y < 1/x

7.∀ x,y,z∈ Q   z + y < z + x  ⇐⇒ x<y
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