
theorem Th53:
  for L being non empty Poset, p being Function of L,L holds (p is
  closure implies Image p is infs-inheriting) & (p is kernel implies Image p is
  sups-inheriting)
proof
  let L be non empty Poset, p be Function of L,L;
  hereby
    assume
A1: p is closure;
    thus Image p is infs-inheriting
    proof
      let X be Subset of Image p;
A2:   the carrier of Image p = rng p by YELLOW_0:def 15;
      then reconsider X9=X as Subset of L by XBOOLE_1:1;
      assume ex_inf_of X,L;
      then p.("/\"(X9,L)) = "/\"(X9,L) by A1,A2,Th28;
      hence thesis by A2,FUNCT_2:4;
    end;
  end;
  assume
A3: p is kernel;
  let X be Subset of Image p;
A4: the carrier of Image p = rng p by YELLOW_0:def 15;
  then reconsider X9=X as Subset of L by XBOOLE_1:1;
  assume ex_sup_of X,L;
  then p.("\/"(X9,L)) = "\/"(X9,L) by A3,A4,Th29;
  hence thesis by A4,FUNCT_2:4;
end;
