
theorem
  for L being non empty Poset, c being Function of L,L st c is closure
  holds corestr c is sups-preserving & for X being Subset of L st X c= the
carrier of Image c & ex_sup_of X,L holds ex_sup_of X,Image c & "\/"(X,Image c)
  = c.("\/"(X,L))
proof
  let L be non empty Poset, c be Function of L,L;
A1: (corestr c) = c by Th30;
  assume
A2: c is closure;
  then
A3: c is idempotent by Def13;
  [inclusion c,corestr c] is Galois by A2,Th36;
  then
A4: corestr c is lower_adjoint;
  hence corestr c is sups-preserving;
  let X be Subset of L such that
A5: X c= the carrier of Image c and
A6: ex_sup_of X,L;
  X c= rng c by A5,YELLOW_0:def 15;
  then
A7: c.:X = X by A3,YELLOW_2:20;
  corestr c preserves_sup_of X by A4,WAYBEL_0:def 33;
  hence thesis by A6,A1,A7;
end;
