
theorem Th12:
  for X being set, p being FinSequence of SmallestPartition X
  ex q being FinSequence of X st product p = {q}
proof
  let X be set;
  set P = SmallestPartition X;
  let p be FinSequence of P;
  set q = the Element of product p;
A1: dom q = dom p by CARD_3:9;
  then reconsider q as FinSequence by Lm1;
A2: q in product p;
A3: product p c= Funcs(dom p, Union p) by FUNCT_6:1;
A4: Union p = union rng p by CARD_3:def 4;
A5: ex f being Function st q = f & dom f = dom p & rng f c= Union p
  by A2,A3,FUNCT_2:def 2;
  Union p c= union P by A4,ZFMISC_1:77;
  then rng q c= union P by A5;
  then rng q c= X by EQREL_1:def 4;
  then reconsider q as FinSequence of X by FINSEQ_1:def 4;
  take q;
  thus product p c= {q}
  proof
    let x be object;
    assume x in product p;
    then consider g being Function such that
A6: x = g and
A7: dom g = dom p and
A8: for x being object st x in dom p holds g.x in p.x by CARD_3:def 5;
    now
      let y be object;
      assume
A9:   y in dom p;
      then
A10:  g.y in p.y by A8;
A11:  q.y in p.y by A9,CARD_3:9;
A12:  p.y in rng p by A9,FUNCT_1:def 3;
      then
A13:  p.y in P;
      reconsider X as non empty set by A12;
      P = the set of all {z} where z is Element of X by EQREL_1:37;
      then consider z being Element of X such that
A14:  p.y = {z} by A13;
      thus g.y = z by A10,A14,TARSKI:def 1
        .= q.y by A11,A14,TARSKI:def 1;
    end;
    then x = q by A1,A6,A7,FUNCT_1:2;
    hence thesis by TARSKI:def 1;
  end;
  thus thesis by ZFMISC_1:31;
end;
