Axiom: Axioms of Connection

For any two distinct points $A,B,$ there is exactly one straight line $l$ such that $A,B$ lie on $l.$ We write $AB=BA=l.$
Any two points $A,B$ which lie on a straight line $l,$ completely determine that straight line, i.e. if $AB=l$ and $AC=l$ and $B\neq C$ then $BC=l.$
There are at least two distinct points that lie on a given straight line. There are at least three distinct points that do not lie on the same straight line.

4 For any three distinct points $A, B, C$ which do not lie on the same straight line, there is exactly one plane $\alpha$ such that $A, B, C$ lie on $\alpha.$ We write $ABC=\alpha.$

Any three distinct points $A, B, C$ which lie on a plane $\alpha$ but do not lie on the same straight line completely determine that plane.
If two distinct points $A, B$ lie both on a line $l$ and a plane $\alpha,$ then all points which lie on $l$ also lie on $\alpha.$ We also say that $l$ is on the plane $\alpha.$
If a point $A$ lies on two distinct planes $\alpha,\beta$ then there is at least another point $B$ which also lies on them.
If a point $A$ lies on two distinct planes $\alpha,\beta$ then there is at least another point $B$ which also lies on them.

Proofs: 1 2 3 4

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