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)
  holds prob(A, B) <= 1
proof
  let E be finite non empty set, A,B be Event of E;
  assume
A1: 0 < prob(B);
  A /\ B c= B by XBOOLE_1:17;
  then prob(A /\ B) * (prob(B))" <= prob(B) * (prob(B))" by A1,Th19,XREAL_1:64;
  then prob(A /\ B) * (prob(B))" <= 1 by A1,XCMPLX_0:def 7;
  hence thesis by XCMPLX_0:def 9;
end;
