let L be lower-bounded continuous LATTICE; :: thesis: for B being join-closed Subset of L st Bottom L in B holds
( B is CLbasis of L iff ( the carrier of () c= B & ( for x, y being Element of L st not y <= x holds
ex b being Element of L st
( b in B & not b <= x & b <= y ) ) ) )

let B be join-closed Subset of L; :: thesis: ( Bottom L in B implies ( B is CLbasis of L iff ( the carrier of () c= B & ( for x, y being Element of L st not y <= x holds
ex b being Element of L st
( b in B & not b <= x & b <= y ) ) ) ) )

assume A1: Bottom L in B ; :: thesis: ( B is CLbasis of L iff ( the carrier of () c= B & ( for x, y being Element of L st not y <= x holds
ex b being Element of L st
( b in B & not b <= x & b <= y ) ) ) )

thus ( B is CLbasis of L implies ( the carrier of () c= B & ( for x, y being Element of L st not y <= x holds
ex b being Element of L st
( b in B & not b <= x & b <= y ) ) ) ) :: thesis: ( the carrier of () c= B & ( for x, y being Element of L st not y <= x holds
ex b being Element of L st
( b in B & not b <= x & b <= y ) ) implies B is CLbasis of L )
proof
assume B is CLbasis of L ; :: thesis: ( the carrier of () c= B & ( for x, y being Element of L st not y <= x holds
ex b being Element of L st
( b in B & not b <= x & b <= y ) ) )

then for x, y being Element of L st x << y holds
ex b being Element of L st
( b in B & x <= b & b << y ) by ;
hence ( the carrier of () c= B & ( for x, y being Element of L st not y <= x holds
ex b being Element of L st
( b in B & not b <= x & b <= y ) ) ) by ; :: thesis: verum
end;
assume that
the carrier of () c= B and
A2: for x, y being Element of L st not y <= x holds
ex b being Element of L st
( b in B & not b <= x & b <= y ) ; :: thesis: B is CLbasis of L
for x, y being Element of L st not y <= x holds
ex b being Element of L st
( b in B & not b <= x & b << y ) by ;
hence B is CLbasis of L by Th46; :: thesis: verum