Ruler postulate:
For every pair of points $P$ and $Q$ there exists a real number PQ, called the distance from $P$ to $Q$. For each line $l$ there is a one-to-one correspondence from $l$ to $R$ such that if $P$ and $Q$ are points on the line that correspond to the real numbers $x$ and $y$, respectively, then $PQ=|x−y|$.
Its implications:
(1) There are infinitely many points lying on a line segment. (provable utilizing axioms of incidence and axioms of order)
(2) There is a one-to-one correspondence between real numbers and points on a line.
(3) Two distinct points determine exactly one line. (provable utilizing axiom I.1 and axiom I.2)
(4) Definition of distance
(5) $PQ=|x−y|$
How can we derive (2), (4) and (5)?