Using only the laws of propositional logic and the rules of inference, prove

that the following proof is valid.

$1.$ If Jack went up the hill and broke his crown, then Jack fell down the hill.

$2.$ If Jack did not fall down the hill, then if Jack didn$$t break his crown, Jill did not

fall down the hill.

$3.$ If Jill did not fall down the hill, Jill fetched a pail of water.

$4.$ Jack went up the hill.

$5.$ Jill did not fetch a pail of water.

$$ Jack and Jill fell down the hil