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(B)
  & A c= B holds prob(A, B) = prob(A) / prob(B)
proof
  let E be finite non empty set, A,B be Event of E;
  assume that
A1: 0 < prob(B) and
A2: A c= B;
  prob(A, B) = 1 - prob(B \ A) / prob(B) by A1,Th36;
  then prob(A, B) = 1 - ( prob(B) - prob(A) ) / prob(B) by A2,Th24;
  then prob(A, B) = 1 - ( prob(B) / prob(B) - prob(A) / prob(B) ) by
XCMPLX_1:120;
  then prob(A, B) = 1 - ( 1 - prob(A) / prob(B) ) by A1,XCMPLX_1:60;
  hence thesis;
end;
