theorem Th3A:
  for A being finite Subset of Union the Sorts of Free(S,Y)
  ex y st for v st v in A holds v is y-omitting
  proof
    let A be finite Subset of Union the Sorts of Free(S,Y);
    set I = union {vf v: v in A};
A0: A is finite;
    deffunc F(Element of Free(S,Y)) = vf $1;
A1: {F(v): v in A} is finite from FRAENKEL:sch 21(A0);
    now let a be set; assume a in {F(v): v in A};
      then ex v st a = F(v) & v in A;
      hence a is finite;
    end;
    then reconsider I as finite set by A1,FINSET_1:7;
    set y = the Element of Y.s \ I;
    reconsider y as Element of Y.s;
    take y;
    let v;
    assume v in A;
    then F(v) in {F(v1): v1 in A};
    then y nin I & F(v) c= I by XBOOLE_0:def 5,ZFMISC_1:74;
    hence v is y-omitting by Th92;
  end;
