reserve N for Cardinal;
reserve M for Aleph;
reserve X for non empty set;
reserve Y,Z,Z1,Z2,Y1,Y2,Y3,Y4 for Subset of X;
reserve S for Subset-Family of X;
reserve x for set;
reserve F,Uf for Filter of X;
reserve S for non empty Subset-Family of X;
reserve I for Ideal of X;
reserve S,S1 for Subset-Family of X;
reserve FS for non empty Subset of Filters(X);
reserve X for infinite set;
reserve Y,Y1,Y2,Z for Subset of X;
reserve F,Uf for Filter of X;
reserve x for Element of X;
reserve X for set;
reserve M for non limit_cardinal Aleph;
reserve F for Filter of M;
reserve N1,N2,N3 for Element of predecessor M;
reserve K1,K2 for Element of M;
reserve T for Inf_Matrix of predecessor M, M, bool M;

theorem Th36:
  for X for N being Cardinal st N c= card X ex Y being set st Y c=
  X & card Y = N
proof
  let X;
  X,card X are_equipotent by CARD_1:def 2;
  then consider f being Function such that
A1: f is one-to-one and
A2: dom f = card X and
A3: rng f = X by WELLORD2:def 4;
  let N be Cardinal;
  assume N c= card X;
  then
A4: dom (f|N) = N by A2,RELAT_1:62;
  take f.:N;
  thus f.:N c= X by A3,RELAT_1:111;
A5: rng (f|N) =f.:N by RELAT_1:115;
  f|N is one-to-one by A1,FUNCT_1:52;
  then N,f.:N are_equipotent by A4,A5,WELLORD2:def 4;
  hence thesis by CARD_1:def 2;
end;
