reserve Y for non empty set;
reserve B for Subset of Y;

theorem
  for a being constant Function of Y,BOOLEAN holds a
  is_dependent_of %O(Y)
proof
  let a be constant Function of Y,BOOLEAN;
  for F being set st F in %O(Y) holds for x1,x2 being set st x1 in F & x2
  in F holds a.x1=a.x2
  proof
    let F be set;
    for x1,x2 being Element of Y holds a.x1=a.x2
    proof
      let x1,x2 be Element of Y;
      per cases by Th20;
      suppose
A1:     a = O_el(Y);
        then a.x1 = FALSE by Def10;
        hence thesis by A1,Def10;
      end;
      suppose
A2:     a = I_el(Y);
        then a.x1 = TRUE by Def11;
        hence thesis by A2,Def11;
      end;
    end;
    hence thesis;
  end;
  hence thesis;
end;
