let A be RelStr ; :: thesis: ( A is with_suprema implies not A is empty )
assume A1: for x, y being Element of A ex z being Element of A st
( x <= z & y <= z & ( for z9 being Element of A st x <= z9 & y <= z9 holds
z <= z9 ) ) ; :: according to LATTICE3:def 10 :: thesis: not A is empty
consider x, y being Element of A;
consider z being Element of A such that
A2: x <= z and
y <= z and
for z9 being Element of A st x <= z9 & y <= z9 holds
z <= z9 by A1;
[x,z] in the InternalRel of A by A2, ORDERS_2:def 9;
hence not A is empty ; :: thesis: verum