The inductive hypothesis therefore applies, and shows that this new collection has nonempty intersection.
2.
The well-ordering of the values of single parameters is provided by the inductive hypothesis of the transfinite induction.
3.
This subgroup contains an element of order " p " by the inductive hypothesis, and we are done.
4.
The Kerala school made intuitive use of mathematical induction, though the inductive hypothesis was not yet formulated or employed in proofs.
5.
In fact, the loop invariant is often the same as the inductive hypothesis to be proved for a recursive program equivalent to a given loop.
6.
This is especially true in inductive proofs, where the given expression is taken to be the inductive hypothesis, and the target expression the inductive conclusion.
7.
By the inductive hypothesis each child has at least 2 ^ { bh ( v')-1 }-1 internal nodes, so v'has at least:
8.
If f ( 1, 1 ), then the inductive hypothesis is that any function g built using only f also has g ( 1, 1 ) = 1.
9.
By the inductive hypothesis the tree rooted at " p " " i " contains at most 4 \ log _ 2 p _ i-4 values, so the entire tree contains at most:
10.
The right hand side of the inequality in Lemma 2 only contains Ramsey numbers for " c " " 1 colours and 2 colours, and therefore exists and is a finite number " t ", by the inductive hypothesis.
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