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 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 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 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 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: ( ( 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: 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 A1, Th47;
hence 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 A1, Lm2; :: thesis: verum

end;

assume 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
then 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 Lm3;
hence B is CLbasis of L by Th46; :: thesis: verum