
theorem Th26:
  for S be non empty finite set,
  D be EqSampleSpaces of S,
  s be Element of D,
  f be Function of S,BOOLEAN
  holds Prob(('not' f),s) = 1 - Prob(f,s)
  proof
    let S be non empty finite set,
    D be EqSampleSpaces of S,
    s be Element of D,
    f be Function of S,BOOLEAN;
    A1: s is Element of S* by FINSEQ_1:def 11;
    reconsider s0 = dom s as finite set;
    reconsider CfS = Coim(f*s,TRUE) as finite set;
    card Seg len s = len s by FINSEQ_1:57; then
    A2: card s0 = len s by FINSEQ_1:def 3;
    A3:Coim(f*s,TRUE) in bool (dom s) by A1,Th18;
    trueEVENT(('not' f)*s)=Coim(( 'not' f)*s,TRUE)
    .=dom s \ Coim (f*s,TRUE) by A1,Th23; then
    A4: card(trueEVENT(('not' f)*s))
    =card(s0) - card(CfS) by A3,CARD_2:44;
    thus Prob(('not' f),s) = card(s0)/(len s)
    - card(CfS)/(len s) by A4,XCMPLX_1:120
    .=1-Prob(f,s) by A2,XCMPLX_1:60;
  end;
