**Proof of Theorem 11**

For each axiom ,as summed up in section 2.2, we prove .

**Rationality**- does not have the form
; instead, we have to show
that for each theorem (and any
*P*in the environment) the formula is a tautology.

**Saying parts of a joint message**-

**Saying contents of an encrypted message**-

**Seeing parts of a joint message**-

**Awareness**-

**Possessing keys of a seen key statement**-

**Possessing believed keys**-

**Decryption**-

**Good key ensures the utterer**- We prove the soundness of this equivalent variant of the
axiom:
,where we may use the assumption that :