theorem
  R linearly_orders Y & X c= Y implies R linearly_orders X
proof
  assume that
A1: R is_reflexive_in Y and
A2: R is_transitive_in Y and
A3: R is_antisymmetric_in Y and
A4: R is_connected_in Y and
A5: X c= Y;
  thus R is_reflexive_in X & R is_transitive_in X & R is_antisymmetric_in X &
  R is_connected_in X by A1,A2,A3,A4,A5;
end;
