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 Th10:
  Y in Tarski-Class(X,succ A) iff
  Y c= Tarski-Class(X,A) & Y in Tarski-Class X or
  ex Z st Z in Tarski-Class(X,A) & (Y c= Z or Y = bool Z)
proof
  set T1 = { u : ex v st v in Tarski-Class(X,A) & u c= v };
  set T2 = { bool v : v in Tarski-Class(X,A) };
  set T3 = bool Tarski-Class(X,A) /\ Tarski-Class X;
A1: Tarski-Class(X,succ A) = T1 \/ T2 \/ T3 by Lm1;
  thus Y in Tarski-Class(X,succ A) implies
  Y c= Tarski-Class(X,A) & Y in Tarski-Class X or
  ex Z st Z in Tarski-Class(X,A) & (Y c= Z or Y = bool Z)
  proof
    assume Y in Tarski-Class(X,succ A);
then A2: Y in T1 \/ T2 or Y in T3 by A1,XBOOLE_0:def 3;
A3: now
      assume Y in T1;
then    ex u st Y = u & ex v st v in Tarski-Class(X,A) & u c= v;
      hence ex Z st Z in Tarski-Class(X,A) & (Y c= Z or Y = bool Z);
    end;
 now
      assume Y in T2;
then    ex v st Y = bool v & v in Tarski-Class(X,A);
      hence ex Z st Z in Tarski-Class(X,A) & (Y c= Z or Y = bool Z);
    end;
   hence thesis by A2,A3,XBOOLE_0:def 3,def 4;
  end;
  assume
A4: Y c= Tarski-Class(X,A) & Y in Tarski-Class X or
  ex Z st Z in Tarski-Class(X,A) & (Y c= Z or Y = bool Z);
A5: now
    assume Y c= Tarski-Class(X,A) & Y in Tarski-Class X;
then  Y in T3 by XBOOLE_0:def 4;
    hence thesis by A1,XBOOLE_0:def 3;
  end;
 now
    given Z such that
A6: Z in Tarski-Class(X,A) and
A7: Y c= Z or Y = bool Z;
    reconsider Z as Element of Tarski-Class X by A6;
    reconsider y = Y as Element of Tarski-Class X by A6,A7,Th3,Th4;
A8: now
      assume Y c= Z;
then   y in T1 by A6;
then   Y in T1 \/ T2 by XBOOLE_0:def 3;
      hence thesis by A1,XBOOLE_0:def 3;
    end;
 now
      assume Y = bool Z;
then   y in T2 by A6;
then   Y in T1 \/ T2 by XBOOLE_0:def 3;
      hence thesis by A1,XBOOLE_0:def 3;
    end;
    hence thesis by A7,A8;
  end;
  hence thesis by A4,A5;
end;
