
theorem Th18:
  for L being RelStr, R being auxiliary(i) (Relation of L), C
being set, x being Element of L st x in C & [x,x] in R & ex_sup_of SetBelow (R,
  C,x),L holds x = sup SetBelow (R,C,x)
proof
  let L be RelStr, R be auxiliary(i) (Relation of L), C be set, x be Element
  of L;
  assume that
A1: x in C and
A2: [x,x] in R and
A3: ex_sup_of SetBelow (R,C,x),L;
A4: for a being Element of L st SetBelow (R,C,x) is_<=_than a holds x <= a
  by A1,A2,Th15;
  SetBelow (R,C,x) is_<=_than x by Th16;
  hence thesis by A4,A3,YELLOW_0:def 9;
end;
