
theorem Th3:
  for X,Y being set st X is_finer_than Y
  for p being FinSequence of X ex q being FinSequence of Y st
  product p c= product q
proof
  let X,Y be set;
  assume
A1: for a being set st a in X ex b being set st b in Y & a c= b;
  let p be FinSequence of X;
  defpred P[object,object] means ex D2 being set st D2 = $2 &  p.$1 c= D2;
A2: for i being object st i in dom p ex y being object st y in Y & P[i,y]
  proof
    let i be object;
    assume
A3:   i in dom p;
     reconsider i as set by TARSKI:1;
     p.i in rng p by FUNCT_1:def 3,A3;
     then p.i in X;
     then consider b being set such that
A4:    b in Y & p.i c= b by A1;
    take b;
    thus thesis by A4;
  end;
  consider q being Function such that
A5: dom q = dom p & rng q c= Y &
for i being object st i in dom p holds P[i,q.i]
  from FUNCT_1:sch 6(A2);
  dom p = Seg len p by FINSEQ_1:def 3;
  then q is FinSequence by A5,FINSEQ_1:def 2;
  then
A6: q is FinSequence of Y by A5,FINSEQ_1:def 4;
  for i being object st i in dom p holds p.i c= q.i
   proof let i being object;
    assume i in dom p;
     then P[i,q.i] by A5;
    hence thesis;
   end;
  then product p c= product q by A5,CARD_3:27;
  hence thesis by A6;
end;
