
theorem
  for X being set holds 'not' FlatCoh X = bool X & 'not' bool X = FlatCoh X
proof
  let X be set;
  thus 'not' FlatCoh X = bool X
  proof
    hereby
      let x be object;
     reconsider xx=x as set by TARSKI:1;
      assume x in 'not' FlatCoh X;
      then xx c= union FlatCoh X by Th65;
      then xx c= X by Th2;
      hence x in bool X;
    end;
    let x be object;
     reconsider xx=x as set by TARSKI:1;
A1: now
      let a be Element of FlatCoh X;
      per cases by Th1;
      suppose
        a = {};
        then xx /\ a c= {0};
        hence ex z being set st xx /\ a c= {z};
      end;
      suppose
        ex z being set st a = {z} & z in X;
        then consider z being set such that
A2:     a = {z} and
        z in X;
        take z;
        thus xx /\ a c= {z} by A2,XBOOLE_1:17;
      end;
    end;
    assume x in bool X;
    then xx c= X;
    then xx c= union FlatCoh X by Th2;
    hence thesis by A1;
  end;
  hence thesis by Th70;
end;
