
theorem Th50:
  for L being non empty Poset, p being Function of L,L st p is
monotone for Lk being Subset of L st Lk = {k where k is Element of L: p.k <= k}
  holds subrelstr Lk is infs-inheriting
proof
  let L be non empty Poset, p be Function of L,L such that
A1: p is monotone;
  let Lk be Subset of L such that
A2: Lk = {k where k is Element of L: p.k <= k};
  let X be Subset of subrelstr Lk;
  assume
A3: ex_inf_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 Lk;
    then x in Lk 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_0:31;
    then x >= "/\"(X,L) by A4;
    then p.x >= p.("/\"(X,L)) by A1;
    hence thesis by A5,ORDERS_2:3;
  end;
  then "/\"(X,L) >= p.("/\"(X,L)) by A3,YELLOW_0:31;
  then "/\"(X,L) in Lk by A2;
  hence thesis by YELLOW_0:def 15;
end;
