
theorem ::Lemma 4.24.iii
  for R being non empty RelStr st R is quasi_ordered & R is connected
  holds <=E R linearly_orders Class(EqRel R)
proof
  let R be non empty RelStr such that
A1: R is quasi_ordered and
A2: R is connected;
A3: <=E R partially_orders Class(EqRel R) by A1,Th8;
  hence <=E R is_reflexive_in Class(EqRel R);
  thus <=E R is_transitive_in Class(EqRel R) by A3;
  thus <=E R is_antisymmetric_in Class(EqRel R) by A3;
  thus <=E R is_connected_in Class(EqRel R)
  proof
    set CR = the carrier of R;
    let x, y be object such that
A4: x in Class(EqRel R) and
A5: y in Class(EqRel R) and x <> y and
A6: not [x,y] in <=E R;
    consider a being object such that
A7: a in CR and
A8: x = Class(EqRel R, a) by A4,EQREL_1:def 3;
    consider b being object such that
A9: b in CR and
A10: y = Class(EqRel R, b) by A5,EQREL_1:def 3;
    reconsider a9=a,b9=b as Element of R by A7,A9;
    not a9 <= b9 by A6,A8,A10,Def5;
    then b9 <= a9 by A2,WAYBEL_0:def 29;
    hence thesis by A8,A10,Def5;
  end;
end;
