reserve X for set;

theorem Th22:
  for S being non empty Subset-Family of X holds (for A being set
  holds A in S implies 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
proof
  let S be non empty Subset-Family of X;
  hereby
    assume that
A1: for A being set holds A in S implies X\A in S and
A2: 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 union M in S
    proof
      let M be N_Sub_set_fam of X;
      assume
A3:   M c= S;
A4:   X \ M c= S
      proof
        let y be object;
        assume
A5:     y in X \ M;
        then reconsider B = y as Subset of X;
        B` in M by A5,SETFAM_1:def 7;
        then B`` in S by A1,A3;
        hence thesis;
      end;
      X \ M is N_Sub_set_fam of X by Th21;
      then X \ meet (X \ M) in S by A1,A2,A4;
      hence thesis by Th4;
    end;
    then reconsider
    S9 = S as non empty compl-closed sigma-additive Subset-Family
    of X by A1,Def1,Def5;
    S9 is SigmaField of X;
    hence S is SigmaField of X;
  end;
  assume
A6: S is SigmaField of X;
  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;
    assume
A7: M c= S;
A8: X \ M c= S
    proof
      let y be object;
      assume
A9:   y in X \ M;
      then reconsider B = y as Subset of X;
      B` in M by A9,SETFAM_1:def 7;
      then B`` in S by A6,A7,PROB_1:def 1;
      hence thesis;
    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 A6,Def1;
    hence thesis by Th4;
  end;
  hence thesis by A6,Def1;
end;
