reserve W,X,Y,Z for set,
  f,g for Function,
  a,x,y,z for set;
reserve u,v for Element of Tarski-Class(X),
  A,B,C for Ordinal,
  L for Sequence;

theorem Th9:
  A <> {} & A is limit_ordinal implies
  Tarski-Class(X,A) = { u : ex B st B in A & u in Tarski-Class(X,B) }
proof
  assume
A1: A <> {} & A is limit_ordinal;
  deffunc f(Ordinal) = Tarski-Class(X,$1);
  consider L such that
A2: dom L = A & for B st B in A holds L.B = f(B) from ORDINAL2:sch 2;
A3: Tarski-Class(X,A) = (union rng L) /\ Tarski-Class X by A1,A2,Lm1;
  thus
  Tarski-Class(X,A) c= { u : ex B st B in A & u in Tarski-Class(X,B) }
  proof
    let x be object;
    assume x in Tarski-Class(X,A);
then  x in union rng L by A3,XBOOLE_0:def 4;
    then consider Y such that
A4: x in Y and
A5: Y in rng L by TARSKI:def 4;
    consider y being object such that
A6: y in dom L and
A7: Y = L.y by A5,FUNCT_1:def 3;
    reconsider y as Ordinal by A6;
 Y = Tarski-Class(X,y) by A2,A6,A7;
    hence thesis by A2,A4,A6;
  end;
  let x be object;
  assume x in { u : ex B st B in A & u in Tarski-Class(X,B) };
  then consider u such that
A8: x = u and
A9: ex B st B in A & u in Tarski-Class(X,B);
  consider B such that
A10: B in A and
A11: u in Tarski-Class(X,B) by A9;
 L.B = Tarski-Class(X,B) by A2,A10;
then  Tarski-Class(X,B) in rng L by A2,A10,FUNCT_1:def 3;
then  u in union rng L by A11,TARSKI:def 4;
  hence thesis by A3,A8,XBOOLE_0:def 4;
end;
