reserve a for set;

theorem Th13:
  for L being non empty RelStr, AR being Relation of L
  for a being object, y being Element of L
   holds a in AR-below y iff [a,y] in AR
proof
  let L be non empty RelStr, AR be Relation of L;
  let a be object, y be Element of L;
  a in AR-below y iff [a,y] in AR
  proof
    hereby
      assume a in AR-below y;
      then ex z being Element of L st ( a = z)&( [z,y] in AR);
      hence [a,y] in AR;
    end;
    assume
A1: [a,y] in AR;
    then reconsider x9 = a as Element of L by ZFMISC_1:87;
    ex z being Element of L st a = z & [z,y] in AR
    proof
      take x9;
      thus thesis by A1;
    end;
    hence thesis;
  end;
  hence thesis;
end;
