
theorem Th24:
  for F being non empty finite set, A being non-empty FinSequence
  of bool F, i, j being Element of NAT, B being Singlification of A, i, C being
Singlification of B, j st i in dom A & j in dom A & C.i <> {} & B.j <> {} holds
  C is Singlification of A, j & C is Singlification of A, i
proof
  let F be non empty finite set, A be non-empty FinSequence of bool F,
      i, j be Element of NAT, B be Singlification of A, i,
      C be Singlification of B, j;
  assume that
A1: i in dom A and
A2: j in dom A and
A3: C.i <> {} and
A4: B.j <> {};
A5: dom B = dom A by Def6;
  then
A6: C.i c= B.i by A1,Def6;
A7: A.i <> {} by A1,Th2;
  then card (B.i) = 1 by A1,Def7;
  then
A8: card (C.i) = 1 by A3,A6,NAT_1:25,43;
A9: A.j <> {} by A2,Th2;
A10: C is Reduction of A by Th22;
  card (C.j) = 1 by A2,A4,A5,Def7;
  hence thesis by A1,A2,A7,A9,A10,A8,Def7;
end;
