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)