
theorem
  for S be non empty finite set,
  BS be Subset of S holds
  ex judgefunc be Function of S,BOOLEAN st
  Coim(judgefunc,TRUE) = BS
  proof
    let S be non empty finite set,
    BS be Subset of S;
    reconsider f = chi(BS,S) as Function of S, BOOLEAN by MARGREL1:def 11;
A1: dom f = S by FUNCT_2:def 1;
A2: for x be object holds x in BS iff x in Coim(f,TRUE)
    proof
      let x be object;
      A3: x in BS implies f.x in {TRUE}
      proof
        assume A4:x in BS;
        f.x = TRUE by A4,FUNCT_3:def 3;
        hence thesis by TARSKI:def 1;
      end;
      f.x in {TRUE} implies x in BS
      proof
        assume f.x in {TRUE};then
        f.x=TRUE by TARSKI:def 1;
        hence thesis by FUNCT_3:36;
      end;
      hence thesis by A3,A1,FUNCT_1:def 7;
    end;
    take f;
    thus thesis by A2,TARSKI:2;
  end;
