
theorem Th9:
  for P being mutually-disjoint set for x being Subset of
  ProdMatroid P ex f being Function of x,P st
  for a being object st a in x holds a in f.a
proof
  defpred P[object,object] means ex D2 being set st D2 = $2 &  $1 in D2;
  let P be mutually-disjoint set;
  let x be Subset of ProdMatroid P;
A1: now
    let a be object;
    assume a in x;
    then a in the carrier of ProdMatroid P;
    then a in union P by Def7;
    then ex y being set st a in y & y in P by TARSKI:def 4;
    hence ex y being object st y in P & P[a,y];
  end;
  consider f being Function of x,P such that
A2: for a being object st a in x holds P[a,f.a] from FUNCT_2:sch 1(A1);
  take f;
  let a be object;
  assume a in x;
   then P[a,f.a] by A2;
  hence thesis;
end;
