
theorem
  for L being non empty Poset, R being auxiliary(i) auxiliary(ii) (
  Relation of L), C being non empty strict_chain of R, X being Subset of C st
ex_inf_of (uparrow "\/"(X,L)) /\ C,L & ex_sup_of X,L & C is maximal holds "\/"(
X,subrelstr C) = "/\"((uparrow "\/"(X,L)) /\ C,L) & (not "\/"(X,L) in C implies
  "\/"(X,L) < "/\"((uparrow "\/"(X,L)) /\ C,L))
proof
  let L be non empty Poset, R be auxiliary(i) auxiliary(ii) (Relation of L), C
  be non empty strict_chain of R, X be Subset of C;
  set s = "\/"(X,L), x = "\/"(X,subrelstr C), U = uparrow s;
  assume that
A1: ex_inf_of U /\ C,L and
A2: ex_sup_of X,L and
A3: C is maximal;
A4: s <= s;
  reconsider x1 = x as Element of L by YELLOW_0:58;
A5: the carrier of subrelstr C = C by YELLOW_0:def 15;
  per cases;
  suppose
A6: s in C;
    then
A7: s = x by A2,A5,YELLOW_0:64;
A8: U /\ C is_>=_than x1
    proof
      let b be Element of L;
      assume b in U /\ C;
      then b in U by XBOOLE_0:def 4;
      hence x1 <= b by A7,WAYBEL_0:18;
    end;
    for a being Element of L st U /\ C is_>=_than a holds a <= x1
    proof
      s in U by A4,WAYBEL_0:18;
      then
A9:   x1 in U /\ C by A6,A7,XBOOLE_0:def 4;
      let a be Element of L;
      assume U /\ C is_>=_than a;
      hence thesis by A9;
    end;
    hence thesis by A1,A6,A8,YELLOW_0:def 10;
  end;
  suppose
    not s in C;
    then consider cs being Element of L such that
A10: cs in C and
A11: s < cs and
A12: not [s,cs] in R and
A13: ex cs1 being Element of subrelstr C st cs = cs1 & X is_<=_than
cs1 & for a being Element of subrelstr C st X is_<=_than a holds cs1 <= a by A2
,A3,Lm2;
A14: s <= cs by A11,ORDERS_2:def 6;
A15: for a being Element of L st U /\ C is_>=_than a holds a <= cs
    proof
      cs in U by A14,WAYBEL_0:18;
      then
A16:  cs in U /\ C by A10,XBOOLE_0:def 4;
      let a be Element of L;
      assume U /\ C is_>=_than a;
      hence thesis by A16;
    end;
A17: cs <= cs;
A18: U /\ C is_>=_than cs
    proof
      let b be Element of L;
      assume
A19:  b in U /\ C;
      then b in U by XBOOLE_0:def 4;
      then
A20:  s <= b by WAYBEL_0:18;
      b in C by A19,XBOOLE_0:def 4;
      then [b,cs] in R or b = cs or [cs,b] in R by A10,Def3;
      hence cs <= b by A12,A17,A20,WAYBEL_4:def 3,def 4;
    end;
    ex_sup_of X,subrelstr C by A2,A3,Th8;
    then cs = x by A13,YELLOW_0:def 9;
    hence thesis by A15,A1,A11,A18,YELLOW_0:def 10;
  end;
end;
