State and prove the principle of inclusion-exclusion. Use it to calculate $\phi(4199)$, where $\phi$ is Euler's $\phi$-function.

In a certain large college, a survey revealed that $90 \%$ of the fellows detest at least one of the pop stars Hairy, Dirty and Screamer. $45 \%$ detest Hairy, $28 \%$ detest Dirty and $46 \%$ detest Screamer. If $27 \%$ detest only Screamer and $6 \%$ detest all three, what proportion detest Hairy and Dirty but not Screamer?