
theorem
  for L being complete non empty Poset, R being extra-order (Relation of
L), C being satisfying_SIC strict_chain of R, a, b being Element of L st a in C
& b in C & a < b ex d being Element of L st d in SupBelow (R,C) & a < d & [d,b]
  in R
proof
  let L be complete non empty Poset, R be extra-order (Relation of L), C be
  satisfying_SIC strict_chain of R, a, b be Element of L;
  assume that
A1: a in C and
A2: b in C and
A3: a < b;
  consider d being Element of L such that
A4: a < d and
A5: [d,b] in R and
A6: d = sup SetBelow (R,C,d) by A1,A2,A3,Th19;
  take d;
  thus thesis by A4,A5,A6,Def10;
end;
