
theorem
  for L be non empty RelStr for S be Subset of L holds S is sups-closed
  iff for X be Subset of S st ex_sup_of X,L holds "\/"(X,L) in S
proof
  let L be non empty RelStr;
  let S be Subset of L;
  thus S is sups-closed implies for X be Subset of S st ex_sup_of X,L holds
  "\/"(X,L) in S
  proof
    assume S is sups-closed;
    then
A1: subrelstr S is sups-inheriting;
    let X be Subset of S;
    assume
A2: ex_sup_of X,L;
    X is Subset of subrelstr S by YELLOW_0:def 15;
    then "\/"(X,L) in the carrier of subrelstr S by A1,A2;
    hence thesis by YELLOW_0:def 15;
  end;
  assume
A3: for X be Subset of S st ex_sup_of X,L holds "\/"(X,L) in S;
  now
    let X be Subset of subrelstr S;
    assume
A4: ex_sup_of X,L;
    X is Subset of S by YELLOW_0:def 15;
    then "\/"(X,L) in S by A3,A4;
    hence "\/"(X,L) in the carrier of subrelstr S by YELLOW_0:def 15;
  end;
  then subrelstr S is sups-inheriting;
  hence thesis;
end;
