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 Th41:
  X c= Y & Y in Rank A implies X in Rank A
proof
  assume that
A1: X c= Y and
A2: Y in Rank A;
A3: now
    given B such that
A4: A = succ B;
A5: Rank succ B = bool Rank B by Lm2;
then  X c= Rank B by A1,A2,A4,XBOOLE_1:1;
    hence thesis by A4,A5;
  end;
 now
    assume
A6: for B holds A <> succ B;
then A7: A is limit_ordinal by ORDINAL1:29;
    then consider B such that
A8: B in A and
A9: Y in Rank B by A2,Lm2,Th31;
 Y c= Rank B by A9,ORDINAL1:def 2;
then A10: X c= Rank B by A1;
A11: bool Rank B = Rank succ B by Lm2;
 succ B in A by A6,A8,ORDINAL1:28,29;
    hence thesis by A7,A10,A11,Th31;
  end;
  hence thesis by A3;
end;
