
theorem Th5:
  for L being non empty lower-bounded antisymmetric RelStr, R being
auxiliary(iv) (Relation of L), C being strict_chain of R holds C \/ {Bottom L}
  is strict_chain of R
proof
  let L be non empty lower-bounded antisymmetric RelStr, R be auxiliary(iv) (
  Relation of L), C be strict_chain of R;
  set A = C \/ {Bottom L};
  let x, y be set;
  assume that
A1: x in A and
A2: y in A;
  reconsider x, y as Element of L by A1,A2;
  per cases by A1,A2,Lm1;
  suppose
    x in C & y in C;
    hence thesis by Def3;
  end;
  suppose
    x in C & y = Bottom L;
    hence thesis by WAYBEL_4:def 6;
  end;
  suppose
    x = Bottom L & y in C;
    hence thesis by WAYBEL_4:def 6;
  end;
  suppose
    x = Bottom L & y = Bottom L;
    hence thesis;
  end;
end;
