
theorem
  for L being non empty transitive RelStr, k being Function of L, L st k
  is sups-preserving holds corestr k is sups-preserving
proof
  let L be non empty transitive RelStr, k be Function of L, L such that
A1: k is sups-preserving;
  let X be Subset of L;
  assume
A2: ex_sup_of X,L;
  set f = corestr k;
A3: k = corestr k by WAYBEL_1:30;
A4: k preserves_sup_of X by A1;
  then
A5: ex_sup_of k.:X, L by A2;
  reconsider fX = f.:X as Subset of Image k;
  dom k = the carrier of L by FUNCT_2:def 1;
  then rng k = the carrier of Image k & k.sup X in rng k by FUNCT_1:def 3
,YELLOW_0:def 15;
  then "\/"(fX, L) is Element of Image k by A4,A3,A2;
  hence ex_sup_of f.:X, Image k by A3,A5,YELLOW_0:64;
  sup (k.:X) = k.sup X by A4,A2;
  hence thesis by A3,A5,YELLOW_0:64;
end;
