
theorem Th19:
  for L being complete non empty Poset, R being extra-order (
Relation of L), C being satisfying_SIC strict_chain of R, p, q being Element of
L st p in C & q in C & p < q ex y being Element of L st p < y & [y,q] in R & y
  = sup SetBelow (R,C,y)
proof
  let L be complete non empty Poset, R be extra-order (Relation of L), C be
  satisfying_SIC strict_chain of R, p, q be Element of L such that
A1: p in C and
A2: q in C and
A3: p < q;
A4: R satisfies_SIC_on C by Def7;
  not q <= p by A3,ORDERS_2:6;
  then not [q,p] in R by WAYBEL_4:def 3;
  then [p,q] in R by A1,A2,A3,Def3;
  then consider w being Element of L such that
A5: w in C and
A6: [p,w] in R and
A7: [w,q] in R and
A8: p < w by A1,A2,A3,A4,Th13;
  consider x1 being Element of L such that
A9: x1 in C and
  [p,x1] in R and
A10: [x1,w] in R and
A11: p < x1 by A1,A4,A5,A6,A8,Th13;
  defpred P[set,set,set] means ex b being Element of L st $3 = b &
  $3 in C & [$2,$3] in R & b <= w;
A12: q <= q;
  reconsider D = SetBelow(R,C,w) as non empty set by A9,A10,Th15;
  reconsider g = x1 as Element of D by A9,A10,Th15;
A13: for n being Nat, x being Element of D ex y being Element of
  D st P[n,x,y]
  proof
    let n be Nat;
    let x be Element of D;
    x in D;
    then reconsider t = x as Element of L;
A14: x in C by Th15;
A15: [x,w] in R by Th15;
    per cases;
    suppose
      x <> w;
      then consider b being Element of L such that
A16:  b in C and
A17:  [x,b] in R and
A18:  [b,w] in R and
      t < b by A4,A5,A14,A15,Th13;
      reconsider y = b as Element of D by A16,A18,Th15;
      take y, b;
      thus thesis by A16,A17,A18,WAYBEL_4:def 3;
    end;
    suppose
A19:  x = w;
      take x, t;
      thus thesis by A19,Th15;
    end;
  end;
  consider f being sequence of  D such that
A20: f.0 = g and
A21: for n being Nat holds P[n,f.n,f.(n+1)] from RECDEF_1:sch 2(A13);
  reconsider f as sequence of  the carrier of L by FUNCT_2:7;
  take y = sup rng f;
A22: ex_sup_of rng f,L by YELLOW_0:17;
A23: dom f = NAT by FUNCT_2:def 1;
  then x1 <= y by A20,A22,FUNCT_1:3,YELLOW_4:1;
  hence p < y by A11,ORDERS_2:7;
  rng f is_<=_than w
  proof
    let x be Element of L;
    assume x in rng f;
    then consider n being object such that
A24: n in dom f and
A25: f.n = x by FUNCT_1:def 3;
    reconsider n as Element of NAT by A24;
A26: ex b being Element of L st f.(n+1) = b & f.(n+1) in C &
    [f.n,f.(n+1)] in R & b <= w by A21;
    then x <= f.In(n+1,NAT) by A25,WAYBEL_4:def 3;
    hence x <= w by A26,ORDERS_2:3;
  end;
  then y <= w by A22,YELLOW_0:def 9;
  hence [y,q] in R by A7,A12,WAYBEL_4:def 4;
A27: ex_sup_of SetBelow (R,C,y),L by YELLOW_0:17;
A28: for x being Element of L st SetBelow (R,C,y) is_<=_than x holds y <= x
  proof
    let x be Element of L such that
A29: SetBelow (R,C,y) is_<=_than x;
    rng f is_<=_than x
    proof
      defpred P[Nat] means f.$1 in C;
      let m be Element of L;
      assume m in rng f;
      then consider n being object such that
A30:  n in dom f and
A31:  f.n = m by FUNCT_1:def 3;
      reconsider n as Element of NAT by A30;
A32:  f.n <= f.n;
A33:  for k being Nat st P[k] holds P[k+1]
      proof
        let k be Nat;
        ex b being Element of L st f.(k+1) = b & f.(k+1) in
        C & [f.k,f.(k+1)] in R & b <= w by A21;
        hence thesis;
      end;
A34:  P[0] by A9,A20;
      for n being Nat holds P[n] from NAT_1:sch 2(A34,A33);
      then
A35:  f.n in C;
A36:  ex b being Element of L st f.(n+1) = b & f.(n+1) in C
      & [f.n,f.(n+1)] in R & b <= w by A21;
      f.In(n+1,NAT) <= y by A22,A23,FUNCT_1:3,YELLOW_4:1;
      then [m,y] in R by A31,A36,A32,WAYBEL_4:def 4;
      then m in SetBelow (R,C,y) by A31,A35,Th15;
      hence m <= x by A29;
    end;
    hence thesis by A22,YELLOW_0:def 9;
  end;
  SetBelow (R,C,y) is_<=_than y by Th16;
  hence thesis by A28,A27,YELLOW_0:def 9;
end;
