
theorem
  for L being lower-bounded non empty Poset, R being extra-order (
Relation of L), C being non empty strict_chain of R st C is sup-closed & (for c
  being Element of L st c in C holds ex_sup_of SetBelow (R,C,c),L) & R
  satisfies_SIC_on C holds for c being Element of L st c in C holds c = sup
  SetBelow (R,C,c)
proof
  let L be lower-bounded non empty Poset, R be extra-order (Relation of L), C
  be non empty strict_chain of R;
  assume that
A1: C is sup-closed and
A2: for c being Element of L st c in C holds ex_sup_of SetBelow (R,C,c), L;
  assume
A3: R satisfies_SIC_on C;
  let c be Element of L such that
A4: c in C;
A5: ex_sup_of SetBelow (R,C,c),L by A2,A4;
  set d = sup SetBelow (R,C,c);
  SetBelow (R,C,c) c= C by XBOOLE_1:17;
  then d = "\/"(SetBelow (R,C,c),subrelstr C) by A1,A5;
  then d in the carrier of subrelstr C;
  then
A6: d in C by YELLOW_0:def 15;
  per cases;
  suppose
    c = d;
    hence thesis;
  end;
  suppose
A7: c <> d;
A8: now
      assume
A9:   c < d;
A10:  for a being Element of L st SetBelow (R,C,c) is_<=_than a holds c <= a
      proof
        let a be Element of L;
        assume SetBelow (R,C,c) is_<=_than a;
        then
A11:    d <= a by A5,YELLOW_0:def 9;
        c <= d by A9,ORDERS_2:def 6;
        hence thesis by A11,ORDERS_2:3;
      end;
      SetBelow (R,C,c) is_<=_than c by Th16;
      hence thesis by A10,A5,YELLOW_0:def 9;
    end;
    [c,d] in R or [d,c] in R by A7,A4,A6,Def3;
    then c <= d or [d,c] in R by WAYBEL_4:def 3;
    then consider y being Element of L such that
A12: y in C and
    [d,y] in R and
A13: [y,c] in R and
A14: d < y by A8,A3,A4,A6,A7,Th13,ORDERS_2:def 6;
    y in SetBelow (R,C,c) by A12,A13,Th15;
    hence thesis by A5,A14,ORDERS_2:6,YELLOW_4:1;
  end;
end;
