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;
reserve k,n,m for Nat;
reserve l for Element of omega;

theorem
  card X in card Y implies Y \ X <> {}
proof
  assume that
A1: card X in card Y and
A2: Y \ X = {};
  Y c= X by A2,XBOOLE_1:37;
  hence contradiction by A1,Th10,ORDINAL1:5;
end;
