reserve A for non empty set;
reserve a,b,c,x,y,z for Element of A;
reserve o,o9 for Element of LinPreorders A;
reserve o99 for Element of LinOrders A;

theorem Th9:
  ex o st for a st a <> b holds a <_o, b
proof
  defpred P[set,set] means $1 <> b or $2 = b;
  consider R being Relation of A such that
A1: for x,y holds [x,y] in R iff P[x,y] from RELSET_1:sch 2;
A2: now
    let x,y;
 P[x,y] or P[y,x];
    hence [x,y] in R or [y,x] in R by A1;
  end;
 now
    let x,y,z;
    assume that
A3: [x,y] in R and
A4: [y,z] in R;
 P[x,y] by A1,A3;
    hence [x,z] in R by A1,A4;
  end;
  then reconsider o = R as Element of LinPreorders A by A2,Def1;
  take o;
  let a;
  assume a <> b;
then  not [b,a] in R by A1;
  hence thesis;
end;
