
theorem Th25:
  for S be non empty finite set,
  D be EqSampleSpaces of S,
  s be Element of D,
  f,g be Function of S,BOOLEAN
  holds Prob((f '&' g),s) =
  card (trueEVENT(f*s) /\ trueEVENT(g*s))/(len s)
  proof
    let S be non empty finite set,
    D be EqSampleSpaces of S,
    s be Element of D,
    f,g be Function of S,BOOLEAN;
    A1: s is Element of S* by FINSEQ_1:def 11;
    trueEVENT((f '&' g)*s) =Coim((f '&' g)*s,TRUE)
    .= Coim (f*s,TRUE) /\ Coim(g*s,TRUE) by Th22,A1
    .= trueEVENT(f*s) /\ trueEVENT(g*s);
    hence thesis;
  end;
