reserve Omega for set;
reserve m,n,k for Nat;
reserve x,y for object;
reserve r,r1,r2,r3 for Real;
reserve seq,seq1 for Real_Sequence;
reserve Sigma for SigmaField of Omega;
reserve ASeq,BSeq for SetSequence of Sigma;
reserve A, B, C, A1, A2, A3 for Event of Sigma;
reserve Omega for non empty set;
reserve Sigma for SigmaField of Omega;
reserve A, B, C, A1, A2, A3 for Event of Sigma;
reserve ASeq,BSeq for SetSequence of Sigma;
reserve P,P1,P2 for Probability of Sigma;

theorem
  for E being Event of Sigma st E = {} holds A, E are_independent_respect_to P
proof
  let E be Event of Sigma;
A1: P.(A /\ ({} Sigma)) = P.A * 0 by VALUED_0:def 19
    .= P.A * P.({} Sigma) by VALUED_0:def 19;
  assume E = {};
  hence thesis by A1;
end;
