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 Th46:
  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`) = 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;
  prob(B) - 1 < 1 - 1 by A2,XREAL_1:9;
  then 0 < - ( - ( 1 - prob(B) ) );
  then
A4: 0 < prob(B`) by Th22;
  then prob(A) * prob(B`, A) = prob(B`) * prob(A, B`) by A1,Th39;
  then prob(A) * 1 = prob(B`) * prob(A, B`) by A1,A3,Th45;
  then prob(A) * (prob(B`))" = prob(A, B`) * ( prob(B`) * (prob(B`))" );
  then
A5: prob(A) * (prob(B`))" = prob(A, B`) * 1 by A4,XCMPLX_0:def 7;
  prob(B`) = 1 - prob(B) by Th22;
  hence thesis by A5,XCMPLX_0:def 9;
end;
