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 Th44:
  aleph 0 = omega
proof
  thus aleph 0 c= omega by Lm1,Th7;
  thus omega c= aleph 0
  proof
    let x be object;
    assume
A1: x in omega;
    then reconsider A = x as Ordinal;
    consider n being Element of omega such that
A2: A = n by A1;
    card succ n c= card omega by Th10,ORDINAL1:21;
    hence thesis by A2,Lm1,ORDINAL1:6;
  end;
end;
