reserve X,Y for set,
  x,x1,x2,y,y1,y2,z for set,
  f,g,h for Function;
reserve M for non empty set;
reserve D for non empty set;
reserve P for Relation;
reserve O for Order of X;
reserve R,P for Relation,
  X,X1,X2,Y,Z,x,y,z,u for set,
  g,h for Function,
  O for Order of X,
  D for non empty set,
  d,d1,d2 for Element of D,
  A1,A2,B for Ordinal,
  L,L1,L2 for Sequence;

theorem
  x in X & x is_superior_of R & X c= field R & R is reflexive implies X
  has_upper_Zorn_property_wrt R
proof
  assume that
A1: x in X and
A2: x is_superior_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;
