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 Th11:
  a <> b & a <> c implies ex o st b <_o, a & c <_o, a &
  (b <_o, c iff b <_o9, c) & (c <_o, b iff c <_o9, b)
proof
  assume
A1: a <> b & a <> c;
  defpred P[Element of A,Element of A] means
  ($1 <> a & $1 <=_o9, $2) or $2 = a;
  consider R being Relation of A such that
A2: for x,y holds [x,y] in R iff P[x,y] from RELSET_1:sch 2;
A3: now
    let x,y;
 P[x,y] or P[y,x] by Th4;
    hence [x,y] in R or [y,x] in R by A2;
  end;
 now
    let x,y,z;
    assume [x,y] in R & [y,z] in R;
then  ( P[x,y])& P[y,z] by A2;
then  P[x,z] by Th5;
    hence [x,z] in R by A2;
  end;
  then reconsider o = R as Element of LinPreorders A by A3,Def1;
  take o;
A4: ( not [a,b] in R)& not [a,c] in R by A1,A2;
A5: not [c,b] in R iff b <_o9, c by A1,A2;
 not [b,c] in R iff c <_o9, b by A1,A2;
  hence thesis by A4,A5;
end;
