
theorem Th8:
  for L being non empty Poset, R being auxiliary(i) auxiliary(ii)
  (Relation of L), C being non empty strict_chain of R, X being Subset of C st
  ex_sup_of X,L & C is maximal holds ex_sup_of X,subrelstr C
proof
  let L be non empty Poset, R be auxiliary(i) auxiliary(ii) (Relation of L), C
  be non empty strict_chain of R, X be Subset of C;
  assume that
A1: ex_sup_of X,L and
A2: C is maximal;
  set s = "\/"(X,L);
  per cases;
  suppose
    s in C;
    hence thesis by A1,Th7;
  end;
  suppose
    not s in C;
    then ex cs being Element of L st cs in C & s < cs & not [s,cs] in R & ex
cs1 being Element of subrelstr C st cs = cs1 & X is_<=_than cs1 & for a being
    Element of subrelstr C st X is_<=_than a holds cs1 <= a by A1,A2,Lm2;
    hence thesis by YELLOW_0:15;
  end;
end;
