
theorem Th16:
  for L be non empty RelStr for S be Subset of L holds S is
join-closed iff for x,y be Element of L st x in S & y in S & ex_sup_of {x,y},L
  holds sup {x,y} in S
proof
  let L be non empty RelStr;
  let S be Subset of L;
  thus S is join-closed implies for x,y be Element of L st x in S & y in S &
  ex_sup_of {x,y},L holds sup {x,y} in S
  proof
    assume S is join-closed;
    then
A1: subrelstr S is join-inheriting;
    let x,y be Element of L;
    assume that
A2: x in S and
A3: y in S and
A4: ex_sup_of {x,y},L;
    the carrier of subrelstr S = S by YELLOW_0:def 15;
    hence thesis by A1,A2,A3,A4;
  end;
  assume
A5: for x,y be Element of L st x in S & y in S & ex_sup_of {x,y},L holds
  sup {x,y} in S;
  now
    let x,y be Element of L;
    assume that
A6: x in the carrier of subrelstr S and
A7: y in the carrier of subrelstr S and
A8: ex_sup_of {x,y},L;
    the carrier of subrelstr S = S by YELLOW_0:def 15;
    hence sup {x,y} in the carrier of subrelstr S by A5,A6,A7,A8;
  end;
  then subrelstr S is join-inheriting;
  hence thesis;
end;
