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 Th7:
  a <> b & b <> c & a <> c implies ex o st a <_o, b & b <_o, c
proof
  assume that
A1: a <> b & b <> c and
A2: a <> c;
  defpred P[set,set] means ($1 = a or $2 <> a) & ($1 <> c or $2 = c);
  consider R being Relation of A such that
A3: for x,y holds [x,y] in R iff P[x,y] from RELSET_1:sch 2;
A4: now
    let x,y;
 P[x,y] or P[y,x] by A2;
    hence [x,y] in R or [y,x] in R by A3;
  end;
 now
    let x,y,z;
    assume [x,y] in R & [y,z] in R;
then  ( P[x,y])& P[y,z] by A3;
    hence [x,z] in R by A3;
  end;
  then reconsider o = R as Element of LinPreorders A by A4,Def1;
  take o;
 ( not [b,a] in R)& not [c,b] in R by A1,A3;
  hence thesis;
end;
