
theorem
  for R being non empty reflexive transitive RelStr, x, y being Element
  of R holds x << y iff downarrow x c= waybelow y
proof
  let R be non empty reflexive transitive RelStr, x, y be Element of R;
  hereby
    assume
A1: x << y;
    thus downarrow x c= waybelow y
    proof
      let z be object;
      assume
A2:   z in downarrow x;
      then reconsider z9 = z as Element of R;
      z9 <= x by A2,WAYBEL_0:17;
      then z9 << y by A1,WAYBEL_3:2;
      hence thesis;
    end;
  end;
  x <= x;
  then
A3: x in downarrow x by WAYBEL_0:17;
  assume downarrow x c= waybelow y;
  hence thesis by A3,WAYBEL_3:7;
end;
