reserve A, B, X, Y for set;

theorem
  for N being sup-Semilattice, A being Subset of N st subrelstr A is
  join-inheriting holds A is directed
proof
  let N be sup-Semilattice, A be Subset of N such that
A1: subrelstr A is join-inheriting;
  let x, y be Element of N such that
A2: x in A & y in A;
  take x"\/"y;
A3: the carrier of subrelstr A = A by YELLOW_0:def 15;
  ex_sup_of {x,y},N by YELLOW_0:20;
  then sup {x,y} in the carrier of subrelstr A by A1,A2,A3;
  hence x"\/"y in A by A3,YELLOW_0:41;
  thus x <= x"\/"y & y <= x"\/"y by YELLOW_0:22;
end;
