
theorem Th1:
  for X being set, F being finite Subset-Family of X ex G being
finite Subset-Family of X st G c= F & union G = union F & for g being Subset of
  X st g in G holds not g c= union (G\{g})
proof
  let X be set;
  defpred P[Nat] means for F being finite Subset-Family of X st card F = $1 ex
  G being finite Subset-Family of X st G c= F & union G = union F & for g being
  Subset of X st g in G holds not g c= union (G\{g});
A1: now
    let n be Nat;
    assume
A2: P[n];
    thus P[n+1]
    proof
      let F be finite Subset-Family of X;
      assume
A3:   card F = n+1;
      per cases;
      suppose
        ex g being Subset of X st g in F & g c= union (F\{g});
        then consider g being Subset of X such that
A4:     g in F and
A5:     g c= union (F\{g});
        reconsider FF = F\{g} as finite Subset-Family of X;
        {g} c= F by A4,ZFMISC_1:31;
        then
A6:     F = FF \/ {g} by XBOOLE_1:45;
A7:     union F c= union FF
        proof
          let x be object;
          assume x in union F;
          then consider X being set such that
A8:       x in X and
A9:       X in F by TARSKI:def 4;
          per cases by A6,A9,XBOOLE_0:def 3;
          suppose
            X in FF;
            then X c= union FF by ZFMISC_1:74;
            hence thesis by A8;
          end;
          suppose
            X in {g};
            then X = g by TARSKI:def 1;
            hence thesis by A5,A8;
          end;
        end;
        g in {g} by TARSKI:def 1;
        then not g in FF by XBOOLE_0:def 5;
        then card (FF \/ {g}) = card FF + 1 by CARD_2:41;
        then consider G being finite Subset-Family of X such that
A10:    G c= FF and
A11:    union G = union FF and
A12:    for g being Subset of X st g in G holds not g c= union (G\{g}
        ) by A2,A3,A6,XCMPLX_1:2;
        take G;
        FF c= F by A6,XBOOLE_1:7;
        hence G c= F by A10;
        union FF c= union F by A6,XBOOLE_1:7,ZFMISC_1:77;
        hence union G = union F by A11,A7;
        thus thesis by A12;
      end;
      suppose
A13:    not ex g being Subset of X st g in F & g c= union (F\{g});
        take G = F;
        thus G c= F;
        thus union G = union F;
        thus thesis by A13;
      end;
    end;
  end;
  let F be finite Subset-Family of X;
A14: card F = card F;
A15: P[0]
  proof
    let F be finite Subset-Family of X;
    assume
A16: card F = 0;
    take G = F;
    thus G c= F;
    thus union G = union F;
    thus thesis by A16;
  end;
  for n being Nat holds P[n] from NAT_1:sch 2(A15, A1);
  hence thesis by A14;
end;
