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

hence waybelow x is join-closed ; :: thesis: verum

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))

then
subrelstr (waybelow x) is join-inheriting
;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;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

hence waybelow x is join-closed ; :: thesis: verum