reserve N for Cardinal;
reserve M for Aleph;
reserve X for non empty set;
reserve Y,Z,Z1,Z2,Y1,Y2,Y3,Y4 for Subset of X;
reserve S for Subset-Family of X;
reserve x for set;
reserve F,Uf for Filter of X;
reserve S for non empty Subset-Family of X;
reserve I for Ideal of X;

theorem Th12:
  S is_multiplicative_with N implies dual S is_additive_with N
proof
  assume
A1: S is_multiplicative_with N;
  deffunc F(Subset of X) = $1`;
  let S1 be non empty set such that
A2: S1 c= dual S and
A3: card S1 in N;
  reconsider S2=S1 as non empty Subset-Family of X by A2,XBOOLE_1:1;
  set S3 = dual S2;
A4: card {F(Y): Y in S2} c= card S2 from TREES_2:sch 2;
  {Y`: Y in S2} = S3 by Th9;
  then
A5: card S3 in N by A3,A4,ORDINAL1:12;
A6: (union S2)` = [#]X \ union S2 .= meet S3 by SETFAM_1:33;
  S3 c= S by A2,SETFAM_1:37;
  then (meet S3)`` in S by A1,A5;
  hence thesis by A6,SETFAM_1:def 7;
end;
