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;
reserve A,C for Ordinal;

theorem Th64:
  for R,X st R partially_orders X & field R = X & X
  has_lower_Zorn_property_wrt R ex x st x is_minimal_in R
proof
  let R,X such that
A1: R partially_orders X and
A2: field R = X and
A3: X has_lower_Zorn_property_wrt R;
  R = R~~;
  then
A4: X has_upper_Zorn_property_wrt R~ by A3,Th51;
  field(R~) = X by A2,RELAT_1:21;
  then consider x such that
A5: x is_maximal_in R~ by A1,A4,Th41,Th63;
  take x;
  thus thesis by A5,Th59;
end;
