
theorem Th7:
  for R being non empty reflexive transitive RelStr, x, y being
  Element of R holds x << y iff uparrow y c= wayabove x
proof
  let R be non empty reflexive transitive RelStr, x, y be Element of R;
  hereby
    assume
A1: x << y;
    thus uparrow y c= wayabove x
    proof
      let z be object;
      assume
A2:   z in uparrow y;
      then reconsider z9 = z as Element of R;
      y <= z9 by A2,WAYBEL_0:18;
      then x << z9 by A1,WAYBEL_3:2;
      hence thesis;
    end;
  end;
  y <= y;
  then
A3: y in uparrow y by WAYBEL_0:18;
  assume uparrow y c= wayabove x;
  hence thesis by A3,WAYBEL_3:8;
end;
