reserve A,B,C for Ordinal,
  K,L,M,N for Cardinal,
  x,y,y1,y2,z,u for object,X,Y,Z,Z1,Z2 for set,
  n for Nat,
  f,f1,g,h for Function,
  Q,R for Relation;
reserve ff for Cardinal-Function;
reserve F,G for Cardinal-Function;
reserve A,B for set;
reserve A,B for Ordinal;

theorem
  not X is finite iff ex Y st Y c= X & card Y = omega
proof
  thus not X is finite implies ex Y st Y c= X & card Y = omega
  proof
    assume not X is finite;
    then not card X in omega;
    then
A1: omega c= card X by CARD_1:4;
    card X,X are_equipotent by CARD_1:def 2;
    then consider f such that
A2: f is one-to-one and
A3: dom f = card X and
A4: rng f = X;
    take Y = f.:(omega);
    thus Y c= X by A4,RELAT_1:111;
    omega,Y are_equipotent
    proof
      take f|(omega);
      thus thesis by A1,A2,A3,FUNCT_1:52,RELAT_1:62,115;
    end;
    hence thesis by CARD_1:def 2;
  end;
  given Y such that
A5: Y c= X and
A6: card Y = omega;
  thus thesis by A5,A6;
end;
