let L be sup-Semilattice; :: thesis: for x being Element of L holds waybelow x is join-closed
let x be Element of L; :: thesis: waybelow x is join-closed

now :: thesis: for y, z being Element of L st y in the carrier of (subrelstr (waybelow x)) & z in the carrier of (subrelstr (waybelow x)) & ex_sup_of {y,z},L holds
sup {y,z} in the carrier of (subrelstr (waybelow x))

let y, z be Element of L; :: thesis: ( y in the carrier of (subrelstr (waybelow x)) & z in the carrier of (subrelstr (waybelow x)) & ex_sup_of {y,z},L implies sup {y,z} in the carrier of (subrelstr (waybelow x)) )
assume that
A1: y in the carrier of (subrelstr (waybelow x)) and
A2: z in the carrier of (subrelstr (waybelow x)) and
ex_sup_of {y,z},L ; :: thesis: sup {y,z} in the carrier of (subrelstr (waybelow x))
z in waybelow x by A2, YELLOW_0:def 15;
then A3: z << x by WAYBEL_3:7;
y in waybelow x by A1, YELLOW_0:def 15;
then y << x by WAYBEL_3:7;
then y "\/" z << x by A3, WAYBEL_3:3;
then y "\/" z in waybelow x by WAYBEL_3:7;
then sup {y,z} in waybelow x by YELLOW_0:41;
hence sup {y,z} in the carrier of (subrelstr (waybelow x)) by YELLOW_0:def 15; :: thesis: verum

end;

then subrelstr (waybelow x) is join-inheriting ;
hence waybelow x is join-closed ; :: thesis: verum