let L be sup-Semilattice; :: thesis: for x being Element of L holds downarrow x is join-closed
let x be Element of L; :: thesis: downarrow x is join-closed
reconsider x1 = x as Element of L ;

now

let y, z be Element of L; :: thesis: ( y in the carrier of (subrelstr (downarrow x)) & z in the carrier of (subrelstr (downarrow x)) & ex_sup_of {y,z},L implies sup {y,z} in the carrier of (subrelstr (downarrow x)) )
assume that
A1: y in the carrier of (subrelstr (downarrow x)) and
A2: z in the carrier of (subrelstr (downarrow x)) and
ex_sup_of {y,z},L ; :: thesis: sup {y,z} in the carrier of (subrelstr (downarrow x))
z in downarrow x by A2, YELLOW_0:def 15;
then A3: z <= x1 by WAYBEL_0:17;
y in downarrow x by A1, YELLOW_0:def 15;
then y <= x1 by WAYBEL_0:17;
then y "\/" z <= x1 by A3, YELLOW_5:9;
then y "\/" z in downarrow x by WAYBEL_0:17;
then sup {y,z} in downarrow x by YELLOW_0:41;
hence sup {y,z} in the carrier of (subrelstr (downarrow x)) by YELLOW_0:def 15; :: thesis: verum

end;

then subrelstr (downarrow x) is join-inheriting by YELLOW_0:def 17;
hence downarrow x is join-closed by Def2; :: thesis: verum