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 Th37:
  A c= B iff Rank A c= Rank B
proof
  thus A c= B implies Rank A c= Rank B
  proof
A1: A c< B iff A c= B & A <> B;
    assume A c= B;
then  Rank A = Rank B or Rank A in Rank B by Th36,A1,ORDINAL1:11;
    hence thesis by ORDINAL1:def 2;
  end;
  assume that
A2: Rank A c= Rank B and
A3: not A c= B;
 B in A by A3,ORDINAL1:16;
  hence contradiction by A2,ORDINAL1:5,Th36;
end;
