
theorem Th32:
  for S be non empty finite set,
  D be EqSampleSpaces of S,
  A,B be set,
  f,g be Function of S,BOOLEAN
  st A c= S & B c= S & A misses B & f = chi(A,S) & g = chi(B,S)
  holds Prob(f '&' g,D) = 0
  proof
    let S be non empty finite set,
    D be EqSampleSpaces of S,
    A,B be set,
    f,g be Function of S,BOOLEAN;
    assume A1: A c= S & B c= S & A misses B & f = chi(A,S) & g = chi(B,S);
    set s = the Element of D;
    A2:Prob(f '&' g,D) = Prob(f '&' g,s) by Def6
    .= card (trueEVENT(f*s) /\ trueEVENT(g*s))/(len s) by Th25;
    trueEVENT(f*s) /\ trueEVENT(g*s) = {}
    proof
      assume trueEVENT(f*s) /\ trueEVENT(g*s) <> {};
      then consider x be object such that
      A3: x in trueEVENT(f*s) /\ trueEVENT(g*s) by XBOOLE_0:def 1;
      A4:trueEVENT(f*s) = s"(trueEVENT(f)) by Th14
      .=s"(f"{TRUE});
      A5:trueEVENT(g*s) = s"(trueEVENT(g)) by Th14
      .=s"(g"{TRUE});
      x in s"(f"{TRUE}) & x in s"(g"{TRUE}) by A4,A5,A3,XBOOLE_0:def 4;
      then x in dom s & s.x in (f"{TRUE}) & s.x in (g"{TRUE})
      by FUNCT_1:def 7;
      then f.(s.x) in {1} & g.(s.x) in {1} by FUNCT_1:def 7;
      then f.(s.x) = 1 & g.(s.x) = 1 by TARSKI:def 1;
      then s.x in A & s.x in B by A1,FUNCT_3:36;
      then s.x in A /\ B by XBOOLE_0:def 4;
      hence contradiction by A1,XBOOLE_0:def 7;
    end;
    hence thesis by A2;
  end;
