
theorem Th30:
  for F being finite set, A being FinSequence of bool F,
      i being Element of NAT st i in dom A & A is Hall holds
    ex G being Singlification of A, i st G is Hall
proof
  let F be finite set, A be FinSequence of bool F, i be Element of NAT such
  that
A1: i in dom A and
A2: A is Hall;
A3: A.i <> {} by A1,A2,Th16,CARD_1:27;
  set n = card (A.i);
A4: n >= 1 by A1,A2,Th16;
  defpred P[Element of NAT] means ex G being Reduction of A st G is Hall &
  card (G.i) = $1;
A5: A is Reduction of A by Th26;
  per cases by A4,XXREAL_0:1;
  suppose
    n = 1;
    then A is Singlification of A, i by A1,A5,Def7,CARD_1:27;
    hence thesis by A2;
  end;
  suppose
A6: n > 1;
A7: for k be Element of NAT st k >= 1 & P[k+1] holds P[k]
    proof
      let k be Element of NAT;
      assume that
A8:   k >= 1 and
A9:   P[k+1];
      consider G being Reduction of A such that
A10:  G is Hall and
A11:  card (G.i) = k+1 by A9;
      1 + 1 <= k + 1 by A8,XREAL_1:6;
      then consider x being set such that
A12:  x in G.i and
A13:  Cut (G, i, x) is Hall by A10,A11,Th29;
      set H = Cut (G,i,x);
A14:  dom G = dom A by Def6;
      then H is Reduction of G by A1,Th21;
      then
A15:  H is Reduction of A by Th22;
      card (H.i) = k + 1 - 1 by A1,A11,A14,A12,Th11
        .= k;
      hence thesis by A13,A15;
    end;
    A is Reduction of A by Th26;
    then
A16: ex n be Element of NAT st n > 1 & P[n] by A2,A6;
    P[1] from Regr2(A16,A7);
    then consider G being Reduction of A such that
A17: G is Hall and
A18: card (G.i) = 1;
    G is Singlification of A, i by A1,A3,A18,Def7;
    hence thesis by A17;
  end;
end;
