
theorem
  for L being complete non empty Poset, R being extra-order (Relation of
L), C being satisfying_SIC strict_chain of R holds R satisfies_SIC_on SupBelow
  (R,C)
proof
  let L be complete non empty Poset, R be extra-order (Relation of L), C be
  satisfying_SIC strict_chain of R;
  let c, d be Element of L;
  assume that
A1: c in SupBelow (R,C) and
A2: d in SupBelow (R,C) and
A3: [c,d] in R and
A4: c <> d;
A5: c <= d by A3,WAYBEL_4:def 3;
  deffunc F(Element of L) = {b where b is Element of L: b in SupBelow(R,C) & [
  b,$1] in R};
A6: d = "\/"(F(d),L) by A2,Th24;
A7: ex_sup_of F(d),L by YELLOW_0:17;
  per cases by A4,A6,A7,YELLOW_0:def 9;
  suppose
    not F(d) is_<=_than c;
    then consider g being Element of L such that
A8: g in F(d) and
A9: not g <= c;
    consider y being Element of L such that
A10: g = y and
A11: y in SupBelow(R,C) and
A12: [y,d] in R by A8;
    reconsider y as Element of L;
    take y;
    thus y in SupBelow(R,C) by A11;
    for c being Element of L holds ex_sup_of SetBelow (R,C,c),L by YELLOW_0:17;
    then SupBelow (R,C) is strict_chain of R by Th22;
    then [c,y] in R or c = y or [y,c] in R by A1,A11,Def3;
    hence [c,y] in R by A9,A10,WAYBEL_4:def 3;
    thus [y,d] in R by A12;
    thus thesis by A9,A10;
  end;
  suppose
    ex g being Element of L st F(d) is_<=_than g & not c <= g;
    then consider g being Element of L such that
A13: F(d) is_<=_than g and
A14: not c <= g;
    d <= g by A6,A7,A13,YELLOW_0:def 9;
    hence thesis by A5,A14,ORDERS_2:3;
  end;
end;
