THEOREM

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[A => B] | [B => C]

PROOF

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Suppose to the contrary...

1 ~[[A => B] | [B => C]]

Premise

Prove: A & ~B & [B & ~C]

Apply Imply-And Rule for each '=>' (previously justified here)

2 ~[~[A & ~B] | [B => C]]

Imply-And, 1

3 ~[~[A & ~B] | ~[B & ~C]]

Imply-And, 2

Apply De Morgan' Law

4 ~~[~~[A & ~B] & ~~[B & ~C]]

DeMorgan, 3

Remove each '~~'

5 ~~[A & ~B] & ~~[B & ~C]

Rem DNeg, 4

6 A & ~B & ~~[B & ~C]

Rem DNeg, 5

7 A & ~B & [B & ~C]

Rem DNeg, 6

Apply Split Rule

8 A

Split, 7

9 ~B

Split, 7

10 B & ~C

Split, 7

11 B

Split, 10

12 ~C

Split, 10

Obtain contradication...

13 B & ~B

Join, 11, 9

Apply Conclusion Rule (Proof by contradiction)

14 ~~[[A => B] | [B => C]]

4 Conclusion, 1

As Required:

15 [A => B] | [B => C]

Rem DNeg, 14