let V be RealUnitarySpace; :: thesis: LattStr(# (Subspaces V),(SubJoin V),(SubMeet V) #) is upper-bounded
set S = LattStr(# (Subspaces V),(SubJoin V),(SubMeet V) #);
ex C being Element of LattStr(# (Subspaces V),(SubJoin V),(SubMeet V) #) st
for A being Element of LattStr(# (Subspaces V),(SubJoin V),(SubMeet V) #) holds
( C "\/" A = C & A "\/" C = C )

proof

reconsider C = (Omega). V as Element of LattStr(# (Subspaces V),(SubJoin V),(SubMeet V) #) by Def3;
take C ; :: thesis: for A being Element of LattStr(# (Subspaces V),(SubJoin V),(SubMeet V) #) holds
( C "\/" A = C & A "\/" C = C )
let A be Element of LattStr(# (Subspaces V),(SubJoin V),(SubMeet V) #); :: thesis: ( C "\/" A = C & A "\/" C = C )
reconsider W = A as Subspace of V by Def3;
thus C "\/" A = (SubJoin V) . (C,A) by LATTICES:def 1
.= ((Omega). V) + W by Def7
.= UNITSTR(# the carrier of V, the ZeroF of V, the U7 of V, the Mult of V, the scalar of V #) by Th11
.= C by RUSUB_1:def 3 ; :: thesis: A "\/" C = C
hence A "\/" C = C ; :: thesis: verum

end;

hence LattStr(# (Subspaces V),(SubJoin V),(SubMeet V) #) is upper-bounded by LATTICES:def 14; :: thesis: verum