
theorem Th10:
  for P being mutually-disjoint set for x being Subset of
ProdMatroid P for f being Function of x,P st
  for a being object st a in x holds a
  in f.a holds x is independent iff f is one-to-one
proof
  let P be mutually-disjoint set, x be Subset of ProdMatroid P;
  let f be Function of x,P;
  assume
A1: for a being object st a in x holds a in f.a;
  hereby
    assume
A2: x is independent;
    thus f is one-to-one
    proof
      let a,b be object;
      assume that
A3:   a in dom f and
A4:   b in dom f;
A5:   f.b in rng f by A4,FUNCT_1:def 3;
      f.a in rng f by A3,FUNCT_1:def 3;
      then reconsider D1 = f.a, D2 = f.b as Element of P by A5;
      a in D1 by A1,A3;
      then
A6:   a in x /\ D1 by A3,XBOOLE_0:def 4;
      consider d2 being Element of D2 such that
A7:   x /\ D2 c= {d2} by A2,Th8;
      b in D2 by A1,A4;
      then b in x /\ D2 by A4,XBOOLE_0:def 4;
      then b = d2 by A7,TARSKI:def 1;
      hence thesis by A7,A6,TARSKI:def 1;
    end;
  end;
  assume
A8: f is one-to-one;
  now
    let D be Element of P;
    set d1 = the Element of D;
    assume
A9: for d being Element of D holds x /\ D c/= {d};
    then x /\ D c/= {d1};
    then consider d2 being object such that
A10: d2 in x /\ D and
    d2 nin {d1};
A11: d2 in D by A10,XBOOLE_0:def 4;
A12: d2 in x by A10,XBOOLE_0:def 4;
    then d2 in f.d2 by A1;
    then
A13: f.d2 meets D by A11,XBOOLE_0:3;
    the carrier of ProdMatroid P = union P by Def7;
    then ex y being set st d2 in y & y in P by A10,TARSKI:def 4;
    then
A14: dom f = x by FUNCT_2:def 1;
    then f.d2 in rng f by A12,FUNCT_1:def 3;
    then
A15: f.d2 = D by A13,TAXONOM2:def 5;
    x /\ D c= {d2}
    proof
      let a be object;
      assume
A16:  a in x /\ D;
      then
A17:  a in x by XBOOLE_0:def 4;
A18:  a in D by A16,XBOOLE_0:def 4;
      a in f.a by A1,A17;
      then
A19:  f.a meets D by A18,XBOOLE_0:3;
      f.a in rng f by A14,A17,FUNCT_1:def 3;
      then f.a = D by A19,TAXONOM2:def 5;
      then a = d2 by A8,A12,A14,A15,A17;
      hence thesis by TARSKI:def 1;
    end;
    hence contradiction by A9,A11;
  end;
  hence thesis by Th8;
end;
