
theorem
  for L being non empty lower-bounded antisymmetric RelStr, R being
auxiliary(iv) (Relation of L), C being strict_chain of R st C is maximal holds
  Bottom L in C
proof
  let L be non empty lower-bounded antisymmetric RelStr, R be auxiliary(iv) (
  Relation of L), C be strict_chain of R such that
A1: for D being strict_chain of R st C c= D holds C = D;
  C \/ {Bottom L} is strict_chain of R by Th5;
  then
A2: C \/ {Bottom L} = C by A1,XBOOLE_1:7;
  assume not Bottom L in C;
  then not Bottom L in {Bottom L} by A2,XBOOLE_0:def 3;
  hence thesis by TARSKI:def 1;
end;
