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 Th10:
  for E,F st E is non empty Subset of Sigma & E is
intersection_stable & F is non empty Subset of Sigma & F is intersection_stable
  holds (for p,q st p in E & q in F holds p,q are_independent_respect_to P)
  implies for u,v st u in sigma(E) & v in sigma(F) holds u,v
  are_independent_respect_to P
proof
  let E,F;
  assume
A1: E is non empty Subset of Sigma;
  assume
A2: E is intersection_stable;
  assume
A3: F is non empty Subset of Sigma;
  assume
A4: F is intersection_stable;
  assume
A5: for p,q st p in E & q in F holds p,q are_independent_respect_to P;
  reconsider F as non empty Subset of Sigma by A3;
  reconsider E as non empty Subset of Sigma by A1;
A6: E c= Indep(F,P) by A5,Th7;
  reconsider E, F as Subset-Family of Omega;
  reconsider sF=sigma(F) as non empty Subset of Sigma by PROB_1:def 9;
  sigma(E) c= Indep(sF,P) by A2,A4,A6,Th9;
  hence thesis by Th7;
end;
