:: deftheorem Def9 defines bool MONOID_1:def 9 :

for G being non empty multMagma holds

( ( G is unital implies bool G = multLoopStr(# (bool the carrier of G),( the multF of G .:^2),{(the_unity_wrt the multF of G)} #) ) & ( not G is unital implies bool G = multMagma(# (bool the carrier of G),( the multF of G .:^2) #) ) );

for G being non empty multMagma holds

( ( G is unital implies bool G = multLoopStr(# (bool the carrier of G),( the multF of G .:^2),{(the_unity_wrt the multF of G)} #) ) & ( not G is unital implies bool G = multMagma(# (bool the carrier of G),( the multF of G .:^2) #) ) );