let A be RelStr ; :: thesis: ( A is with_infima implies not A is empty )
assume A3: for x, y being Element of A ex z being Element of A st
( x >= z & y >= z & ( for z' being Element of A st x >= z' & y >= z' holds
z >= z' ) ) ; :: according to LATTICE3:def 11 :: thesis: not A is empty
consider x, y being Element of A;
consider z being Element of A such that
A4: ( x >= z & y >= z ) and
for z' being Element of A st x >= z' & y >= z' holds
z >= z' by A3;
[z,x] in the InternalRel of A by A4, ORDERS_2:def 9;
then x in the carrier of A by ZFMISC_1:106;
hence not A is empty ; :: thesis: verum