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Supplement to Common Knowledge

() is common knowledge for the agents of

**Proof**.

Since is a coarsening of _{i} for each i N, **K**_{ i}( ( ) ).
Hence, **K**^{ 1}_{N} ( ( ) ),
and since by definition **K**_{ i}( ( ) ) = { | _{i}( ) ( ) } = ( ),

K^{ 1}_{N}( ( ) ) =

^{i N}K_{ i}( ( ) ) = ( )

Applying the recursive definition of mutual knowledge, for any m 1,

K^{ m}_{N}( ( ) ) =

^{i N}K_{ i}(K^{ m - 1}_{N}( ( ) ) =

^{i N}K_{ i}( ( ) ) = ( )

so, since ( )
, by definition we have **K** *_{N} ( ( ) ).

*First published: August 27, 2001*

*Content last modified: August 27, 2001*