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;
