
theorem Th10:
  for S be non empty finite set,
  D be EqSampleSpaces of S,
  a be Element of S,
  s be Element of D,
  judgefunc be Function of S,BOOLEAN
  st (for x be set holds x=a iff judgefunc.x = TRUE) holds
  Prob(judgefunc,s)= FDprobability (a,s)
  proof
    let S be non empty finite set,
    D be EqSampleSpaces of S,
    a be Element of S,
    s be Element of D,
    judgefunc be Function of S,BOOLEAN;
    assume A1:for x be set holds x=a iff judgefunc.x =TRUE;
    A2: for n be set holds n in s"{a} iff n in dom s &s.n =a
    proof
      let n be set;
      n in s"{a} iff n in dom s &s.n in {a} by FUNCT_1:def 7;
      hence thesis by TARSKI:def 1;
    end;
    A3: for x be object holds
    x in (judgefunc*s)"{TRUE} implies x in s"{a}
    proof
      let x be object;
      assume x in (judgefunc*s)"{TRUE};
      then A4:x in dom(judgefunc*s) &
      (judgefunc*s).x in {TRUE} by FUNCT_1:def 7;
      then (judgefunc*s).x = TRUE by TARSKI:def 1;
      then A5:x in dom s &
      judgefunc.(s.x)= TRUE by A4,FUNCT_1:11,12;
      then s.x=a by A1;
      then s.x in {a} by TARSKI:def 1;
      hence thesis by A5,FUNCT_1:def 7;
    end;
    for x be object holds
    x in s"{a} implies x in (judgefunc*s)"{TRUE}
    proof
      let x be object;
      assume A6:x in s"{a};
      then A7: x in dom s &s.x =a by A2;
      s.x in S by A7;
      then A8:s.x in dom judgefunc by FUNCT_2:def 1;
      judgefunc.(s.x)= TRUE by A6,A2,A1;
      then (judgefunc*s).x = TRUE by A6,A2,FUNCT_1:13;
      then A9:(judgefunc*s).x in {TRUE} by TARSKI:def 1;
      x in dom (judgefunc*s) by A7,A8,FUNCT_1:11;
      hence thesis by A9,FUNCT_1:def 7;
    end;
    hence thesis by A3,TARSKI:2;
  end;
