
theorem Th5:
  for L be complete sup-Semilattice for S be join-inheriting non
empty full SubRelStr of L st Bottom L in the carrier of S for X be Subset of L
  for Y be Subset of S st X = Y holds finsups Y c= finsups X
proof
  let L be complete sup-Semilattice;
  let S be join-inheriting non empty full SubRelStr of L;
  assume
A1: Bottom L in the carrier of S;
  let X be Subset of L;
  let Y be Subset of S;
  assume
A2: X = Y;
  let x be object;
  assume x in finsups Y;
  then x in {"\/"(V,S) where V is finite Subset of Y: ex_sup_of V,S} by
WAYBEL_0:def 27;
  then consider Z be finite Subset of Y such that
A3: x = "\/"(Z,S) and
A4: ex_sup_of Z,S;
  reconsider Z as finite Subset of X by A2;
  now
    per cases;
    suppose
      Z is non empty;
      then reconsider Z1 = Z as finite non empty Subset of S by XBOOLE_1:1;
      reconsider xl = "\/"(Z1,L) as Element of S by WAYBEL21:15;
A5:   ex_sup_of Z1,L by YELLOW_0:17;
A6:   now
        let b be Element of S;
        reconsider b1 = b as Element of L by YELLOW_0:58;
        assume
A7:     b is_>=_than Z1;
        b1 is_>=_than Z1
        by A7,YELLOW_0:59;
        then "\/"(Z1, L) <= b1 by A5,YELLOW_0:30;
        hence xl <= b by YELLOW_0:60;
      end;
A8:   "\/"(Z1, L) is_>=_than Z1 by A5,YELLOW_0:30;
      xl is_>=_than Z1
      proof
        let b be Element of S;
        reconsider b1 = b as Element of L by YELLOW_0:58;
        assume b in Z1;
        then b1 <= "\/"(Z1, L) by A8;
        hence b <= xl by YELLOW_0:60;
      end;
      then "\/"(Z1,S) = "\/"(Z1,L) by A6,YELLOW_0:30;
      then x in { "\/"(V,L) where V is finite Subset of X: ex_sup_of V,L } by
A3,A5;
      hence thesis by WAYBEL_0:def 27;
    end;
    suppose
A9:  Z is empty;
      reconsider dL = Bottom L as Element of S by A1;
      reconsider dS = Bottom S as Element of L by YELLOW_0:58;
      S is lower-bounded by A4,A9,WAYBEL20:6;
      then Bottom S <= dL by YELLOW_0:44;
      then
A10:  dS <= Bottom L by YELLOW_0:59;
A11:  ex_sup_of Z,L by YELLOW_0:17;
      Bottom L <= dS by YELLOW_0:44;
      then dS = Bottom L by A10,YELLOW_0:def 3;
      then x in { "\/"(V,L) where V is finite Subset of X: ex_sup_of V,L } by
A3,A9,A11;
      hence thesis by WAYBEL_0:def 27;
    end;
  end;
  hence thesis;
end;
