reserve X for set;

theorem
  for S being Field_Subset of X holds S = X \ S
proof
  let S be Field_Subset of X;
  for A being object holds A in S iff A in X \ S
  proof
    let A be object;
    hereby
      assume
A1:   A in S;
      then reconsider B = A as Subset of X;
      B` in S by A1,PROB_1:def 1;
      hence A in X \ S by SETFAM_1:def 7;
    end;
    assume
A2: A in X \ S;
    then reconsider B = A as Subset of X;
    B` in S by A2,SETFAM_1:def 7;
    then B`` in S by PROB_1:def 1;
    hence thesis;
  end;
  hence thesis by TARSKI:2;
end;
