reserve a for set;

theorem
  for L being sup-Semilattice, AR being Relation of L
  for y being Element of L holds a in AR-above y iff [y,a] in AR
proof
  let L be with_suprema Poset, AR be Relation of L;
  let y be Element of L;
  a in AR-above y iff [y,a] in AR
  proof
    hereby
      assume a in AR-above y;
      then ex z being Element of L st ( a = z)&( [y,z] in AR);
      hence [y,a] in AR;
    end;
    assume
A1: [y,a] in AR;
    then reconsider x9 = a as Element of L by ZFMISC_1:87;
    ex z being Element of L st a = z & [y,z] in AR
    proof
      take x9;
      thus thesis by A1;
    end;
    hence thesis;
  end;
  hence thesis;
end;
