theorem
  x in X & x is_inferior_of R & X c= field R & R is reflexive implies X
  has_lower_Zorn_property_wrt R
proof
  assume that
A1: x in X and
A2: x is_inferior_of R and
A3: X c= field R and
A4: R is_reflexive_in field R;
  let Y such that
A5: Y c= X and
  R|_2 Y is being_linear-order;
  take x;
  thus x in X by A1;
  let y;
  assume y in Y;
  then
A6: y in X by A5;
  y = x or y <> x;
  hence thesis by A2,A3,A4,A6;
end;
