
theorem Th49:
  for L being non empty Poset, p being Function of L,L st p is
monotone for Lc being Subset of L st Lc = {c where c is Element of L: c <= p.c}
  holds subrelstr Lc is sups-inheriting
proof
  let L be non empty Poset, p be Function of L,L such that
A1: p is monotone;
  let Lc be Subset of L such that
A2: Lc = {c where c is Element of L: c <= p.c};
  let X be Subset of subrelstr Lc;
  assume
A3: ex_sup_of X,L;
  p.("\/"(X,L)) is_>=_than X
  proof
    let x be Element of L;
    assume
A4: x in X;
    then x in the carrier of subrelstr Lc;
    then x in Lc by YELLOW_0:def 15;
    then
A5: ex l being Element of L st x = l & l <= p.l by A2;
    ("\/"(X,L)) is_>=_than X by A3,YELLOW_0:30;
    then x <= "\/"(X,L) by A4;
    then p.x <= p.("\/"(X,L)) by A1;
    hence x <= p.("\/"(X,L)) by A5,ORDERS_2:3;
  end;
  then "\/"(X,L) <= p.("\/"(X,L)) by A3,YELLOW_0:30;
  then "\/"(X,L) in Lc by A2;
  hence thesis by YELLOW_0:def 15;
end;
