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 )

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

( 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
for x, y being Element of L st not y <= x holds
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;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

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