
theorem
  for T being lower-bounded sup-Semilattice
  for S being join-inheriting full non empty SubRelStr of T
  st Bottom T in the carrier of S & S is directed-sups-inheriting
  holds S is sups-inheriting
proof
  let T be lower-bounded sup-Semilattice;
  let S be join-inheriting full non empty SubRelStr of T such that
A1: Bottom T in the carrier of S and
A2: S is directed-sups-inheriting;
  let A be Subset of S;
  the carrier of S c= the carrier of T by YELLOW_0:def 13;
  then reconsider C = A as Subset of T by XBOOLE_1:1;
  set F = finsups C;
  assume
A3: ex_sup_of A, T;
  then
A4: sup F = sup C by WAYBEL_0:55;
  F c= the carrier of S
  proof
    let x be object;
    assume x in F;
    then consider Y being finite Subset of C such that
A5: x = "\/"(Y, T) and ex_sup_of Y, T;
    reconsider Y as finite Subset of T by XBOOLE_1:1;
    reconsider Z = Y as finite Subset of S by XBOOLE_1:1;
    assume
A6: not x in the carrier of S;
    then Z <> {} by A1,A5;
    hence thesis by A5,A6,Th15;
  end;
  then reconsider G = F as non empty Subset of S;
  reconsider G as directed non empty Subset of S by WAYBEL10:23;
A7: now
    let Y be finite Subset of C;
    Y c= the carrier of T by XBOOLE_1:1;
    hence Y <> {} implies ex_sup_of Y,T by YELLOW_0:54;
  end;
A8: now
    let x be Element of T;
    assume x in F;
    then ex Y being finite Subset of C st x = "\/"(Y,T) & ex_sup_of Y,T;
    hence ex Y being finite Subset of C st ex_sup_of Y,T & x = "\/"(Y,T);
  end;
  now
    let Y be finite Subset of C;
    reconsider Z = Y as finite Subset of T by XBOOLE_1:1;
    assume Y <> {};
    then ex_sup_of Z, T by YELLOW_0:54;
    hence "\/"(Y,T) in F;
  end;
  then ex_sup_of G, T by A3,A7,A8,WAYBEL_0:53;
  hence thesis by A2,A4;
end;
