reserve E for non empty set;
reserve a for Element of E;
reserve A, B for Subset of E;
reserve Y for set;
reserve p for FinSequence;
reserve e, e1, e2 for Singleton of E;

theorem
  for E being finite non empty set, A,B being Event of E st 0 < prob(A)
  & prob(B) < 1 & A misses B holds prob(A`, B`) = 1 - prob(A) / (1 - prob(B))
proof
  let E be finite non empty set, A,B be Event of E;
  assume that
A1: 0 < prob(A) and
A2: prob(B) < 1 and
A3: A misses B;
A4: prob(B`) = 1 - prob(B) by Th22;
  prob(B) -1 < 1 - 1 by A2,XREAL_1:9;
  then 0 < - ( - ( 1 - prob(B) ) );
  then prob(A` , B`) = 1 - prob(A, B`) by A4,Th40;
  hence thesis by A1,A2,A3,Th46;
end;
