We define counting on by one and give it the special prime symbol.
We define addition by two rules for counting on.
Counting on by zero rule
n+0 = n
Counting on by one more rule
m+n’ = (m+n)’
This says that counting on by one more of the right addend equals counting on by one more from the previous sum.
2+0 = 2
Suppose we have already done
2+3 = 5.
Now we want to do 2+4. Here 4 = 3′ we also know.
2+4 = 2+3′ = (2+3)’ = 5′ = 6
This is how we use that 2+3 = 5 in the substitution step.
If the rules are too difficult just call them facts.
Counting on Facts