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

let x be Element of L; :: thesis: waybelow x is meet-closed

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

let x be Element of L; :: thesis: waybelow x is meet-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_inf_of {y,z},L holds

inf {y,z} in the carrier of (subrelstr (waybelow x))

then
subrelstr (waybelow x) is meet-inheriting
;inf {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_inf_of {y,z},L implies inf {y,z} in the carrier of (subrelstr (waybelow x)) )

assume that

A1: y in the carrier of (subrelstr (waybelow x)) and

z in the carrier of (subrelstr (waybelow x)) and

ex_inf_of {y,z},L ; :: thesis: inf {y,z} in the carrier of (subrelstr (waybelow x))

y in waybelow x by A1, YELLOW_0:def 15;

then A2: y << x by WAYBEL_3:7;

y "/\" z <= y by YELLOW_0:23;

then y "/\" z << x by A2, WAYBEL_3:2;

then y "/\" z in waybelow x by WAYBEL_3:7;

then inf {y,z} in waybelow x by YELLOW_0:40;

hence inf {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

z in the carrier of (subrelstr (waybelow x)) and

ex_inf_of {y,z},L ; :: thesis: inf {y,z} in the carrier of (subrelstr (waybelow x))

y in waybelow x by A1, YELLOW_0:def 15;

then A2: y << x by WAYBEL_3:7;

y "/\" z <= y by YELLOW_0:23;

then y "/\" z << x by A2, WAYBEL_3:2;

then y "/\" z in waybelow x by WAYBEL_3:7;

then inf {y,z} in waybelow x by YELLOW_0:40;

hence inf {y,z} in the carrier of (subrelstr (waybelow x)) by YELLOW_0:def 15; :: thesis: verum

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