
theorem Th24:
  for L being complete non empty Poset, R being extra-order (
Relation of L), C being satisfying_SIC strict_chain of R, d being Element of L
st d in SupBelow (R,C) holds d = "\/"({b where b is Element of L: b in SupBelow
  (R,C) & [b,d] in R},L)
proof
  let L be complete non empty Poset, R be extra-order (Relation of L), C be
  satisfying_SIC strict_chain of R, d be Element of L;
  deffunc F(Element of L) = {b where b is Element of L: b in SupBelow(R,C) & [
  b,$1] in R};
  set p = "\/"(F(d),L);
A1: ex_sup_of SetBelow (R,C,d),L by YELLOW_0:17;
A2: F(d) is_<=_than d
  proof
    let a be Element of L;
    assume a in F(d);
    then ex b being Element of L st a = b & b in SupBelow(R,C) & [b,d] in R;
    hence a <= d by WAYBEL_4:def 3;
  end;
  assume d in SupBelow (R,C);
  then
A3: d = sup SetBelow(R,C,d) by Def10;
  assume
A4: p <> d;
  ex_sup_of F(d),L by YELLOW_0:17;
  then p <= d by A2,YELLOW_0:def 9;
  then
A5: p < d by A4,ORDERS_2:def 6;
  now
    per cases by A3,A1,A4,YELLOW_0:def 9;
    suppose
      not SetBelow(R,C,d) is_<=_than p;
      then consider a being Element of L such that
A6:   a in SetBelow(R,C,d) and
A7:   not a <= p;
A8:   [a,d] in R by A6,Th15;
      a in C by A6,Th15;
      hence ex a being Element of L st a in C & [a,d] in R & not a <= p by A8
,A7;
    end;
    suppose
      ex a being Element of L st SetBelow(R,C,d) is_<=_than a & not p <= a;
      then consider a being Element of L such that
A9:   SetBelow(R,C,d) is_<=_than a and
A10:  not p <= a;
      d <= a by A3,A1,A9,YELLOW_0:def 9;
      then p < a by A5,ORDERS_2:7;
      hence ex a being Element of L st a in C & [a,d] in R & not a <= p by A10,
ORDERS_2:def 6;
    end;
  end;
  then consider cc being Element of L such that
A11: cc in C and
A12: [cc,d] in R and
A13: not cc <= p;
  per cases;
  suppose
    [cc,cc] in R;
    then cc = sup SetBelow (R,C,cc) by A11,Th18,YELLOW_0:17;
    then cc in SupBelow (R,C) by Def10;
    then cc in F(d) by A12;
    hence contradiction by A13,YELLOW_0:17,YELLOW_4:1;
  end;
  suppose
A14: not [cc,cc] in R;
    ex cs being Element of L st cs in C & cc < cs & [cs,d] in R
    proof
      per cases by A3,A1,A12,A14,YELLOW_0:def 9;
      suppose
        not SetBelow(R,C,d) is_<=_than cc;
        then consider cs being Element of L such that
A15:    cs in SetBelow(R,C,d) and
A16:    not cs <= cc;
        take cs;
A17:    not [cs,cc] in R by A16,WAYBEL_4:def 3;
        thus cs in C by A15,Th15;
        then [cc,cs] in R by A17,A11,A16,Def3;
        then cc <= cs by WAYBEL_4:def 3;
        hence cc < cs by A16,ORDERS_2:def 6;
        thus thesis by A15,Th15;
      end;
      suppose
A18:    ex a being Element of L st SetBelow(R,C,d) is_<=_than a & not cc <= a;
        cc in SetBelow(R,C,d) by A11,A12,Th15;
        hence thesis by A18;
      end;
    end;
    then consider cs being Element of L such that
A19: cs in C and
A20: cc < cs and
A21: [cs,d] in R;
    consider y being Element of L such that
A22: cc < y and
A23: [y,cs] in R and
A24: y = sup SetBelow (R,C,y) by A11,A19,A20,Th19;
A25: y in SupBelow (R,C) by A24,Def10;
A26: d <= d;
    y <= cs by A23,WAYBEL_4:def 3;
    then [y,d] in R by A21,A26,WAYBEL_4:def 4;
    then y in F(d) by A25;
    then y <= p by YELLOW_0:17,YELLOW_4:1;
    then cc < p by A22,ORDERS_2:7;
    hence contradiction by A13,ORDERS_2:def 6;
  end;
end;
