reserve Omega, I for non empty set;
reserve Sigma for SigmaField of Omega;
reserve P for Probability of Sigma;
reserve D, E, F for Subset-Family of Omega;
reserve  B, sB for non empty Subset of Sigma;
reserve b for Element of B;
reserve a for Element of Sigma;
reserve p, q, u, v for Event of Sigma;
reserve n, m for Element of NAT;
reserve S, S9, X, x, y, z, i, j for set;

theorem Th9:
  for A being Subset-Family of Omega st A is non empty Subset of
Sigma & A is intersection_stable for B being non empty Subset of Sigma st B is
intersection_stable holds A c= Indep(B,P) implies for D,sB st D=B & sigma(D)=sB
  holds sigma(A) c= Indep(sB,P)
proof
  let A be Subset-Family of Omega;
  assume A is non empty Subset of Sigma;
  then reconsider sA=sigma(A) as non empty Subset of Sigma by PROB_1:def 9;
  assume
A1: A is intersection_stable;
  let B be non empty Subset of Sigma;
  assume
A2: B is intersection_stable;
  assume A c= Indep(B,P);
  then
A3: sigma(A) c= Indep(B,P) by A1,Th6;
  let D, sB;
  assume
A4: D=B & sigma(D)=sB;
  reconsider B as Subset-Family of Omega by XBOOLE_1:1;
  B c= Indep(sA,P) by A3,Th8;
  then sigma(B) c= Indep(sA,P) by A2,Th6;
  hence thesis by A4,Th8;
end;
