
theorem Th28:
  for L being non empty Poset, c being Function of L,L, X being
Subset of L st c is closure & ex_inf_of X,L & X c= rng c holds inf X = c.(inf X
  )
proof
  let L be non empty Poset, c be Function of L,L, X be Subset of L such that
A1: c is projection and
A2: id(L) <= c and
A3: ex_inf_of X,L and
A4: X c= rng c;
A5: c is monotone by A1;
A6: c is idempotent by A1;
  c.(inf X) is_<=_than X
  proof
    let x be Element of L;
    assume
A7: x in X;
    inf X is_<=_than X by A3,YELLOW_0:31;
    then inf X <= x by A7;
    then c.(inf X) <= c.x by A5;
    hence thesis by A4,A6,A7,Lm2;
  end;
  then
A8: c.(inf X) <= inf X by A3,YELLOW_0:31;
  id(L).(inf X) <= c.(inf X) by A2,YELLOW_2:9;
  then inf X <= c.(inf X);
  hence thesis by A8,ORDERS_2:2;
end;
