
theorem Th13:
  for S be non empty finite set,
  f be S-valued Function,
  judgefunc be Function of S,BOOLEAN,
  n be set st n in dom f holds
  n in trueEVENT(judgefunc*f) iff f.n in trueEVENT(judgefunc)
  proof
    let S be non empty finite set,
    f be S-valued Function,
    judgefunc be Function of S,BOOLEAN,
    n be set;
    assume A1: n in dom f;
    A2: trueEVENT(judgefunc*f) is Subset of dom f by Th8;
    thus n in trueEVENT(judgefunc*f) implies
    f.n in trueEVENT(judgefunc)
    proof
      assume A3: n in trueEVENT(judgefunc*f);
      then (judgefunc*f).n in {TRUE} by FUNCT_1:def 7;
      then A4:(judgefunc*f).n = TRUE by TARSKI:def 1;
      judgefunc.(f.n)=TRUE by A4,A3,A2,FUNCT_1:13;
      then A5:judgefunc.(f.n) in {TRUE} by TARSKI:def 1;
      f.n in rng f by A3,A2,FUNCT_1:def 3;
      then f.n in S;
      then f.n in dom judgefunc by FUNCT_2:def 1;
      hence thesis by A5,FUNCT_1:def 7;
    end;
    assume A6:f.n in trueEVENT(judgefunc);
    A7:f.n in dom judgefunc &
    judgefunc.(f.n) in {TRUE} by A6,FUNCT_1:def 7;
    A8:(judgefunc*f).n in {TRUE} by A7,A1,FUNCT_1:13;
    n in dom (judgefunc*f) by A1,A6,FUNCT_1:11;
    hence thesis by A8,FUNCT_1:def 7;
  end;
