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;
