
theorem Th3:
  for X being set, F being Subset-Family of X holds F,COMPLEMENT F
  are_equipotent
proof
  let X be set, F be Subset-Family of X;
  deffunc F(set) = X \ $1;
  consider f being Function such that
A1: dom f = F and
A2: for x being set st x in F holds f.x = F(x) from FUNCT_1:sch 5;
  take f;
  thus f is one-to-one
  proof
    let x1,x2 be object such that
A3: x1 in dom f and
A4: x2 in dom f and
A5: f.x1 = f.x2;
    reconsider X1 = x1, X2 = x2 as Subset of X by A1,A3,A4;
    X1` = f.x1 by A1,A2,A3
      .= X2` by A1,A2,A4,A5;
    hence x1 = X2`` .= x2;
  end;
  thus dom f = F by A1;
  thus rng f c= COMPLEMENT F
  proof
    let e be object;
    assume e in rng f;
    then consider u being object such that
A6: u in dom f and
A7: e = f.u by FUNCT_1:def 3;
    reconsider Y = u as Subset of X by A1,A6;
    e = Y` by A1,A2,A6,A7;
    hence thesis by A1,A6,SETFAM_1:35;
  end;
  let e be object;
  assume
A8: e in COMPLEMENT F;
  then reconsider Y = e as Subset of X;
A9: Y` in F by A8,SETFAM_1:def 7;
  e = Y`` .= f.Y` by A2,A9;
  hence thesis by A1,A9,FUNCT_1:def 3;
end;
