
theorem
  for L being complete non empty Poset, R being auxiliary(i)
  auxiliary(ii) (Relation of L), C being non empty strict_chain of R st C is
  maximal holds subrelstr C is complete
proof
  let L be complete non empty Poset, R be auxiliary(i) auxiliary(ii) (Relation
  of L), C be non empty strict_chain of R;
  assume
A1: C is maximal;
  for X being Subset of subrelstr C holds ex_sup_of X,subrelstr C
  proof
    let X be Subset of subrelstr C;
    X is Subset of C by YELLOW_0:def 15;
    hence thesis by A1,Th8,YELLOW_0:17;
  end;
  hence thesis by YELLOW_0:53;
end;
