set M = the carrier of L;
for u being set st u in the carrier of L holds
u in the carrier of L ;
then reconsider M = the carrier of L as Subset of L by TARSKI:def 3;
reconsider M = M as non empty Subset of L ;
take M ; :: thesis: ( M is add-closed & M is left-ideal & M is right-ideal )
A1: for p, x being Element of L st x in M holds
x * p in M ;
( ( for x, y being Element of L st x in M & y in M holds
x + y in M ) & ( for p, x being Element of L st x in M holds
p * x in M ) ) ;
hence ( M is add-closed & M is left-ideal & M is right-ideal ) by A1, Def1, Def2, Def3; :: thesis: verum