
theorem
  for T being _Tree, X being finite set st for x being set st x in X ex
  t being _Subtree of T st x = the_Vertices_of t holds X is with_Helly_property
proof
  let T be _Tree, X be finite set such that
A1: for x being set st x in X ex t being _Subtree of T st x =
  the_Vertices_of t;
  defpred P[Nat] means for H being non empty finite set st card H = $1 & H c=
  X & for x, y being set st x in H & y in H holds x meets y holds meet H <> {};
A2: for k being Nat st for n being Nat st n < k holds P[n] holds P[k]
  proof
    let k be Nat such that
A3: for n being Nat st n < k holds P[n];
    let H be non empty finite set such that
A4: card H = k and
A5: H c= X and
A6: for x, y being set st x in H & y in H holds x meets y;
    per cases by NAT_1:25;
    suppose
      k = 0;
      hence thesis by A4;
    end;
    suppose
      k = 1;
      then consider x being Element of H such that
A7:   H = { x } by A4,PRE_CIRC:25;
      ex t being _Subtree of T st x = the_Vertices_of t by A1,A5;
      hence thesis by A7,SETFAM_1:10;
    end;
    suppose
A8:   k > 1;
      set cH = the Element of H;
      set A = H \ {cH};
A9:   card A = card H - card {cH} by EULER_1:4
        .= k - 1 by A4,CARD_1:30;
      k-1 > 1-1 by A8,XREAL_1:9;
      then reconsider A as non empty finite set by A9;
A10:  A c= X by A5;
      for x, y being set st x in A & y in A holds x meets y by A6;
      then reconsider mA = meet A as non empty set by A3,A9,A10,XREAL_1:44;
      set cA = the Element of A;
      set B = H \ {cA};
A11:  cA in A;
      then
A12:  card B = card H - card {cA} by EULER_1:4
        .= k - 1 by A4,CARD_1:30;
      set C = {cH, cA};
A13:  meet C = cH /\ cA by SETFAM_1:11;
      cH meets cA by A6,A11;
      then reconsider mC = meet C as non empty set by A13;
      k-1 > 1-1 by A8,XREAL_1:9;
      then reconsider B as non empty finite set by A12;
A14:  B c= X by A5;
      for x, y being set st x in B & y in B holds x meets y by A6;
      then reconsider mB = meet B as non empty set by A3,A12,A14,XREAL_1:44;
      set a = the Element of  mA, b = the Element of  mB,
          c = the Element of  mC;
      c in mC & mC c= union C by SETFAM_1:2;
      then consider cc being set such that
A15:  c in cc and
A16:  cc in C by TARSKI:def 4;
      cH in H;
      then C c= X by A5,A11,A10,ZFMISC_1:32;
      then
A17:  ex cct being _Subtree of T st cc = the_Vertices_of cct by A1,A16;
      a in mA & mA c= union A by SETFAM_1:2;
      then consider aa being set such that
A18:  a in aa and
A19:  aa in A by TARSKI:def 4;
      b in mB & mB c= union B by SETFAM_1:2;
      then consider bb being set such that
A20:  b in bb and
A21:  bb in B by TARSKI:def 4;
A22:  ex bbt being _Subtree of T st bb = the_Vertices_of bbt by A1,A14,A21;
      ex aat being _Subtree of T st aa = the_Vertices_of aat by A1,A10,A19;
      then reconsider a, b, c as Vertex of T by A18,A20,A22,A15,A17;
A23:  cA <> cH by ZFMISC_1:56;
      now
        let s be set;
        assume
A24:    s in H;
        hence ex t being _Subtree of T st s = the_Vertices_of t by A1,A5;
        thus a in s & b in s or a in s & c in s or b in s & c in s
        proof
          per cases;
          suppose
            s = cH;
            then s in C & s in B by A23,TARSKI:def 2,ZFMISC_1:56;
            hence thesis by SETFAM_1:def 1;
          end;
          suppose
A25:        s = cA;
            then s in C by TARSKI:def 2;
            hence thesis by A25,SETFAM_1:def 1;
          end;
          suppose
            s <> cH & s <> cA;
            then s in A & s in B by A24,ZFMISC_1:56;
            hence thesis by SETFAM_1:def 1;
          end;
        end;
      end;
      hence thesis by Th53;
    end;
  end;
A26: for n being Nat holds P[n] from NAT_1:sch 4 (A2);
  let H be non empty set such that
A27: H c= X and
A28: for x, y being set st x in H & y in H holds x meets y;
  reconsider H9 = H as finite set by A27;
  card H9 = card H9;
  hence thesis by A26,A27,A28;
end;
