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 Th14:
  for F being ManySortedSigmaField of I,Sigma, J,K being non empty
Subset of I st F is_independent_wrt P & J misses K holds for u,v st u in sigUn(
  F,J) & v in sigUn(F,K) holds P.(u /\ v) = P.u * P.v
proof
  let F be ManySortedSigmaField of I,Sigma, J,K be non empty Subset of I;
A1: MeetSections(J,F) is non empty Subset of Sigma & MeetSections(K,F) is
  non empty Subset of Sigma by Th13;
  assume
A2: F is_independent_wrt P & J misses K;
A3: for p,q st p in MeetSections(J,F) & q in MeetSections(K,F) holds p,q
  are_independent_respect_to P
  proof
    let p,q;
    assume
A4: p in MeetSections(J,F) & q in MeetSections(K,F);
    reconsider p,q as Subset of Omega;
    P.(p /\ q) = P.p * P.q by A2,A4,Th12;
    hence thesis by PROB_2:def 4;
  end;
  let u,v;
  assume u in sigUn(F,J) & v in sigUn(F,K);
  then u in sigma(MeetSections(J,F)) & v in sigma(MeetSections (K,F)) by Th11;
  then u,v are_independent_respect_to P by A1,A3,Th10;
  hence thesis by PROB_2:def 4;
end;
