
theorem Th8:
  for X being non empty set for P being a_partition of X holds
  the set of all product p where p is Element of P* is a_partition of X*
proof
  let X be non empty set;
  let P be a_partition of X;
  set PP = the set of all product p where p is Element of P*;
  PP c= bool (X*)
  proof
    let x be object;
     reconsider xx=x as set by TARSKI:1;
    assume x in PP;
    then consider p being Element of P* such that
A1: x = product p;
    xx c= X*
    proof
      let y be object;
      assume y in xx;
      then consider f being Function such that
A2:   y = f and
A3:   dom f = dom p and
A4:   for z being object st z in dom p holds f.z in p.z by A1,CARD_3:def 5;
      dom p = Seg len p by FINSEQ_1:def 3;
      then
A5:   y is FinSequence by A2,A3,FINSEQ_1:def 2;
      rng f c= X
      proof
        let z be object;
        assume z in rng f;
        then consider v being object such that
A6:     v in dom p and
A7:     z = f.v by A3,FUNCT_1:def 3;
        p.v in rng p by A6,FUNCT_1:def 3;
        then
A8:     p.v in P;
        z in p.v by A4,A6,A7;
        hence thesis by A8;
      end;
      then y is FinSequence of X by A2,A5,FINSEQ_1:def 4;
      hence thesis by FINSEQ_1:def 11;
    end;
    hence thesis;
  end;
  then reconsider PP as Subset-Family of X*;
  PP is a_partition of X*
  proof
    thus union PP c= X*;
    thus X* c= union PP
    proof
      let x be object;
      assume x in X*;
      then reconsider x as FinSequence of X by FINSEQ_1:def 11;
A9:   rng x c= X;
      union P = X by EQREL_1:def 4;
      then consider p being Function such that
A10:  dom p = dom x and
A11:  rng p c= P and
A12:  x in product p by A9,Th5;
      dom p = Seg len x by A10,FINSEQ_1:def 3;
      then reconsider p as FinSequence by FINSEQ_1:def 2;
      reconsider p as FinSequence of P by A11,FINSEQ_1:def 4;
      reconsider p as Element of P* by FINSEQ_1:def 11;
      product p in PP;
      hence thesis by A12,TARSKI:def 4;
    end;
    let A be Subset of X*;
    assume A in PP;
    then consider p being Element of P* such that
A13: A = product p;
    thus A <> {} by A13;
    let B be Subset of X*;
    assume B in PP;
    then consider q being Element of P* such that
A14: B = product q;
    assume
A15: A <> B;
    assume A meets B;
    then consider x being object such that
A16: x in A and
A17: x in B by XBOOLE_0:3;
    consider f being Function such that
A18: x = f and
A19: dom f = dom p and
A20: for z being object st z in dom p holds f.z in p.z
by A13,A16,CARD_3:def 5;
A21: ex g being Function st ( x = g)&( dom g = dom q)&( for z
    being object st z in dom q holds g.z in q.z) by A14,A17,CARD_3:def 5;
    now
      let z be object;
      assume
A22:  z in dom p;
      then
A23:  f.z in p.z by A20;
A24:  f.z in q.z by A18,A19,A21,A22;
A25:  p.z in rng p by A22,FUNCT_1:def 3;
A26:  q.z in rng q by A18,A19,A21,A22,FUNCT_1:def 3;
A27:  p.z meets q.z by A23,A24,XBOOLE_0:3;
A28:  p.z in P by A25;
      q.z in P by A26;
      hence p.z = q.z by A27,A28,EQREL_1:def 4;
    end;
    hence contradiction by A13,A14,A15,A18,A19,A21,FUNCT_1:2;
  end;
  hence thesis;
end;
