
theorem Th8:
  for P being set for A being Subset of ProdMatroid P holds A is
independent iff for D being Element of P ex d being Element of D st A /\ D c= {
  d}
proof
  let P be set;
  set M = ProdMatroid P;
A1: the_family_of M = {A where A is Subset of union P: for D being set st D
  in P ex d being set st A /\ D c= {d}} by Def7;
  let A be Subset of ProdMatroid P;
A2: the carrier of M = union P by Def7;
  thus A is independent implies for D being Element of P ex d being Element of
  D st A /\ D c= {d}
  proof
    assume A in the_family_of M;
    then
A3: ex B being Subset of union P st A = B & for D being set st D in P ex d
    being set st B /\ D c= {d} by A1;
    let D be Element of P;
    P = {} implies A = {} & {} /\ D = {} by A2,ZFMISC_1:2;
    then P = {} implies A /\ D c= {1};
    then consider d being set such that
A4: A /\ D c= {d} by A3;
    set d0 = the Element of D;
    now
      assume d nin D;
      then d nin A /\ D by XBOOLE_0:def 4;
      then A /\ D <> {d} by TARSKI:def 1;
      then A /\ D = {} by A4,ZFMISC_1:33;
      hence A /\ D c= {d0};
    end;
    hence thesis by A4;
  end;
  assume
A5: for D being Element of P ex d being Element of D st A /\ D c= {d};
A6: now
    let D be set;
    assume D in P;
    then ex d being Element of D st A /\ D c= {d} by A5;
    hence ex d being set st A /\ D c= {d};
  end;
  the carrier of M = union P by Def7;
  hence A in the_family_of M by A1,A6;
end;
