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 Th51:
  X has_upper_Zorn_property_wrt R iff X has_lower_Zorn_property_wrt R~
proof
  thus X has_upper_Zorn_property_wrt R implies X has_lower_Zorn_property_wrt R
  ~
  proof
    assume
A1: for Y st Y c= X & R|_2 Y is being_linear-order ex x st x in X &
    for y st y in Y holds [y,x] in R;
    let Y;
A2: (R|_2 Y)~~ = R|_2 Y;
    assume that
A3: Y c= X and
A4: R~|_2 Y is being_linear-order;
    R~|_2 Y = (R|_2 Y)~ by Lm16;
    then consider x such that
A5: x in X and
A6: for y st y in Y holds [y,x] in R by A1,A2,A3,A4,Th18;
    take x;
    thus x in X by A5;
    let y;
    assume y in Y;
    then [y,x] in R by A6;
    hence thesis by RELAT_1:def 7;
  end;
  assume
A7: for Y st Y c= X & R~|_2 Y is being_linear-order ex x st x in X &
  for y st y in Y holds [x,y] in R~;
  let Y;
  assume that
A8: Y c= X and
A9: R|_2 Y is being_linear-order;
  R~|_2 Y = (R|_2 Y)~ by Lm16;
  then consider x such that
A10: x in X and
A11: for y st y in Y holds [x,y] in R~ by A7,A8,A9,Th18;
  take x;
  thus x in X by A10;
  let y;
  assume y in Y;
  then [x,y] in R~ by A11;
  hence thesis by RELAT_1:def 7;
end;
