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
  for X st X <> {} & for Z st Z <> {} & Z c= X & Z is c=-linear holds
  meet Z in X ex Y st Y in X & for Z st Z in X & Z <> Y holds not Z c= Y
proof
  let X such that
A1: X <> {} and
A2: for Z st Z <> {} & Z c= X & Z is c=-linear holds meet Z in X;
  for Z st Z c= X & Z is c=-linear ex Y st Y in X & for X1 st X1 in Z
  holds Y c= X1
  proof
    set x = the Element of X;
    let Z such that
A3: Z c= X and
A4: Z is c=-linear;
    Z <> {} or Z = {};
    then consider Y such that
A5: Y = meet Z & Z <> {} or Y = x & Z = {};
    take Y;
    thus thesis by A1,A2,A3,A4,A5,SETFAM_1:3;
  end;
  hence thesis by A1,Th66;
end;
