
theorem Th27:
  for F being non empty finite set,
      A being non empty non-empty FinSequence of bool F,
      f being Function holds f is Singlification of A iff
    (dom f = dom A &
    for i being Element of NAT st i in dom A holds
      f is Singlification of A, i)
proof
  let F be non empty finite set, A be non empty non-empty FinSequence of bool
  F, f be Function;
  hereby
    assume f is Singlification of A;
    then reconsider f9 = f as Singlification of A;
    f9 is Reduction of A;
    hence dom f = dom A by Def6;
    let i be Element of NAT;
    assume
A1: i in dom A;
    then card (f9.i) = 1 & A.i <> {} by Def8;
    hence f is Singlification of A, i by A1,Def7;
  end;
  assume that
A2: dom f = dom A and
A3: for i being Element of NAT st i in dom A holds f is Singlification of A, i;
  reconsider f as FinSequence of bool F by A3,FINSEQ_5:6;
  for i being Element of NAT st i in dom A holds f.i c= A.i
  proof
    let i be Element of NAT;
    assume
A4: i in dom A;
    then f is Singlification of A, i by A3;
    hence thesis by A4,Def6;
  end;
  then reconsider f9 = f as Reduction of A by A2,Def6;
  for i being Element of NAT st i in dom A holds card (f9.i) = 1
  proof
    let i be Element of NAT;
    assume
A5: i in dom A;
    then f is Singlification of A, i & A.i <> {} by A3;
    hence thesis by A5,Def7;
  end;
  hence thesis by Def8;
end;
