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 Th8:
  for A,B being non empty Subset of Sigma st A c= Indep(B,P) holds
  B c= Indep(A,P)
proof
  let A, B be non empty Subset of Sigma;
  assume
A1: A c= Indep(B,P);
  for q,p st q in B & p in A holds q,p are_independent_respect_to P
      by A1,Th7,PROB_2:19;
  hence thesis by Th7;
end;
