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
set x = the Element of A;
consider z being Element of A such that
A2: the Element of A <= z and
the Element of A <= z and
for z9 being Element of A st the Element of A <= z9 & the Element of A <= z9 holds
z <= z9 by A1;
[ the Element of A,z] in the InternalRel of A by A2;
hence not A is empty ; :: thesis: verum