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 Th7:
  for A, B being non empty Subset of Sigma holds A c= Indep(B,P)
  iff for p,q st p in A & q in B holds p,q are_independent_respect_to P
proof
  let A, B be non empty Subset of Sigma;
A1: now
    assume
A2: for p,q st p in A & q in B holds p,q are_independent_respect_to P;
    thus A c= Indep(B,P)
    proof
      let x be object;
      assume
A3:   x in A;
      then reconsider x as Event of Sigma;
      for b being Element of B holds P.(x /\ b) = P.x * P.b
             by A2,A3,PROB_2:def 4;
      hence thesis by Def1;
    end;
  end;
  now
    assume
A4: A c= Indep(B,P);
    thus for p,q st p in A & q in B holds p,q are_independent_respect_to P
    proof
      let p,q;
      assume that
A5:   p in A and
A6:   q in B;
      reconsider q as Element of B by A6;
      reconsider p as Element of Sigma;
      P.(p /\ q) = P.p * P.q by A4,A5,Def1;
      hence thesis by PROB_2:def 4;
    end;
  end;
  hence thesis by A1;
end;
