reserve
  a,b for object, I,J for set, f for Function, R for Relation,
  i,j,n for Nat, m for (Element of NAT),
  S for non empty non void ManySortedSign,
  s,s1,s2 for SortSymbol of S,
  o for OperSymbol of S,
  X for non-empty ManySortedSet of the carrier of S,
  x,x1,x2 for (Element of X.s), x11 for (Element of X.s1),
  T for all_vars_including inheriting_operations free_in_itself
  (X,S)-terms MSAlgebra over S,
  g for Translation of Free(S,X),s1,s2,
  h for Endomorphism of Free(S,X);
reserve
  r,r1,r2 for (Element of T),
  t,t1,t2 for (Element of Free(S,X));
reserve
  Y for infinite-yielding ManySortedSet of the carrier of S,
  y,y1 for (Element of Y.s), y11 for (Element of Y.s1),
  Q for all_vars_including inheriting_operations free_in_itself
  (Y,S)-terms MSAlgebra over S,
  q,q1 for (Element of Args(o,Free(S,Y))),
  u,u1,u2 for (Element of Q),
  v,v1,v2 for (Element of Free(S,Y)),
  Z for non-trivial ManySortedSet of the carrier of S,
  z,z1 for (Element of Z.s),
  l,l1 for (Element of Free(S,Z)),
  R for all_vars_including inheriting_operations free_in_itself
  (Z,S)-terms MSAlgebra over S,
  k,k1 for Element of Args(o,Free(S,Z));
reserve c,c1,c2 for set, d,d1 for DecoratedTree;
reserve
  w for (Element of Args(o,T)),
  p,p1 for Element of Args(o,Free(S,X));
reserve C for (context of x), C1 for (context of y), C9 for (context of z),
  C11 for (context of x11), C12 for (context of y11), D for context of s,X;
reserve
  S9 for sufficiently_rich non empty non void ManySortedSign,
  s9 for SortSymbol of S9,
  o9 for s9-dependent OperSymbol of S9,
  X9 for non-trivial ManySortedSet of the carrier of S9,
  x9 for (Element of X9.s9);
reserve h1 for x-constant Homomorphism of Free(S,X), T,
  h2 for y-constant Homomorphism of Free(S,Y), Q;

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;
