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

theorem
  for a being Function of Y,BOOLEAN,PA being a_partition of Y
  holds B_INF(a,PA) '<' a
proof
  let a be Function of Y,BOOLEAN;
  let PA be a_partition of Y;
  B_INF(a,PA) 'imp' a = I_el Y
  proof
    let y be Element of Y;
    per cases;
    suppose
A1:   for x being Element of Y st x in EqClass(y,PA) holds a.x=TRUE;
A2:   a.y = TRUE by A1,EQREL_1:def 6;
      'not' B_INF(a,PA).y = ('not' B_INF(a,PA)).y by MARGREL1:def 19;
      then
      'not' (B_INF(a,PA)).y 'or' a.y = ('not' B_INF(a,PA)).y 'or' (I_el Y)
      .y by A2,Def11
        .= ('not' B_INF(a,PA) 'or' I_el Y).y by Def4
        .= (I_el Y).y by Th9;
      hence thesis by Def8;
    end;
    suppose
      not (for x being Element of Y st x in EqClass(y,PA) holds a.x= TRUE );
      then (B_INF(a,PA)).y = FALSE by Def16;
      then 'not' (B_INF(a,PA)).y 'or' a.y = (I_el Y).y 'or' a.y by Def11
        .= ((I_el Y) 'or' a).y by Def4
        .= (I_el Y).y by Th9;
      hence thesis by Def8;
    end;
  end;
  hence thesis by Th15;
end;
