
theorem Th14:
  for S be non empty finite set,
  f be S-valued Function,
  judgefunc be Function of S,BOOLEAN holds
  trueEVENT(judgefunc*f) = f"(trueEVENT(judgefunc))
  proof
    let S be non empty finite set,
    f be S-valued Function,
    judgefunc be Function of S,BOOLEAN;
    A1: trueEVENT(judgefunc*f) is Subset of dom f by Th8;
    for x be object holds
    x in f"(trueEVENT(judgefunc)) iff x in trueEVENT(judgefunc*f)
    proof
      let x be object;
      thus x in f"(trueEVENT(judgefunc)) implies x in trueEVENT(judgefunc*f)
      proof
        assume x in f"(trueEVENT(judgefunc));then
        x in dom f & f.x in trueEVENT(judgefunc) by FUNCT_1:def 7;
        hence thesis by Th13;
      end;
      assume A2: x in trueEVENT(judgefunc*f);
      f.x in trueEVENT(judgefunc) by Th13,A2,A1;
      hence thesis by A1,A2,FUNCT_1:def 7;
    end;
    hence thesis by TARSKI:2;
  end;
