SE 500
Theorems on Integer Ranges

Theorem IR0: m≤i<m+1 = i=m

Theorem IR1:
(a) m≤r≤s≤n ⇒ (m≤i≤s ∨ r≤i≤n  ≡  m≤i≤n)
(b) m≤r≤s≤n ⇒ (m≤i<s ∨ r≤i<n  ≡  m≤i<n)
(c) m≤r≤s≤n ⇒ (m<i≤s ∨ r<i≤n  ≡  m<i≤n)

For each of the parts of the Theorem above, the special case r=s generates a corollary:

Corollary 1:
(a) m≤r≤n ⇒ (m≤i≤r ∨ r≤i≤n  ≡  m≤i≤n)
(b) m≤r≤n ⇒ (m≤i<r ∨ r≤i<n  ≡  m≤i<n)
(c) m≤r≤n ⇒ (m<i≤r ∨ r<i≤n  ≡  m<i≤n)

Theorem IR2: m≤n ⇒ ((m≤i<j≤n) ∨ (m≤j≤n ∧ i=j)  ≡  m≤i≤j≤n)