let X be set ; :: thesis: for S being non empty Subset-Family of X holds
( ( ( for A being set st A in S holds
X \ A in S ) & ( for M being N_Sub_set_fam of X st M c= S holds
meet M in S ) ) iff S is SigmaField of X )

let S be non empty Subset-Family of X; :: thesis: ( ( ( for A being set st A in S holds
X \ A in S ) & ( for M being N_Sub_set_fam of X st M c= S holds
meet M in S ) ) iff S is SigmaField of X )

hereby :: thesis: ( S is SigmaField of X implies ( ( for A being set st A in S holds
X \ A in S ) & ( for M being N_Sub_set_fam of X st M c= S holds
meet M in S ) ) )
assume that
A1: for A being set st A in S holds
X \ A in S and
A2: for M being N_Sub_set_fam of X st M c= S holds
meet M in S ; :: thesis: S is SigmaField of X
for M being N_Sub_set_fam of X st M c= S holds
union M in S
proof
let M be N_Sub_set_fam of X; :: thesis: ( M c= S implies union M in S )
assume A3: M c= S ; :: thesis: union M in S
A4: X \ M c= S
proof
let y be object ; :: according to TARSKI:def 3 :: thesis: ( not y in X \ M or y in S )
assume A5: y in X \ M ; :: thesis: y in S
then reconsider B = y as Subset of X ;
B ` in M by ;
then (B `) ` in S by A1, A3;
hence y in S ; :: thesis: verum
end;
X \ M is N_Sub_set_fam of X by Th21;
then X \ (meet (X \ M)) in S by A1, A2, A4;
hence union M in S by Th4; :: thesis: verum
end;
then reconsider S9 = S as non empty compl-closed sigma-additive Subset-Family of X by ;
S9 is SigmaField of X ;
hence S is SigmaField of X ; :: thesis: verum
end;
assume A6: S is SigmaField of X ; :: thesis: ( ( for A being set st A in S holds
X \ A in S ) & ( for M being N_Sub_set_fam of X st M c= S holds
meet M in S ) )

for M being N_Sub_set_fam of X st M c= S holds
meet M in S
proof
let M be N_Sub_set_fam of X; :: thesis: ( M c= S implies meet M in S )
assume A7: M c= S ; :: thesis: meet M in S
A8: X \ M c= S
proof
let y be object ; :: according to TARSKI:def 3 :: thesis: ( not y in X \ M or y in S )
assume A9: y in X \ M ; :: thesis: y in S
then reconsider B = y as Subset of X ;
B ` in M by ;
then (B `) ` in S by ;
hence y in S ; :: thesis: verum
end;
X \ M is N_Sub_set_fam of X by Th21;
then union (X \ M) in S by A6, A8, Def5;
then X \ (union (X \ M)) in S by ;
hence meet M in S by Th4; :: thesis: verum
end;
hence ( ( for A being set st A in S holds
X \ A in S ) & ( for M being N_Sub_set_fam of X st M c= S holds
meet M in S ) ) by ; :: thesis: verum