reserve A,B,C for Ordinal,
  X,X1,Y,Y1,Z for set,a,b,b1,b2,x,y,z for object,
  R for Relation,
  f,g,h for Function,
  k,m,n for Nat;
reserve M,N for Cardinal;
reserve S for Sequence;

theorem
  A c= B iff aleph A c= aleph B
proof
A1: aleph A c< aleph B iff aleph A <> aleph B & aleph A c= aleph B;
  A in B iff aleph A in aleph B by Th20;
  hence thesis by A1,Th21,ORDINAL1:11,XBOOLE_0:def 8;
end;
