
theorem Th12:
  for L being non empty reflexive transitive RelStr
  for x,y being Element of L st x <= y
  holds waybelow x c= waybelow y & wayabove y c= wayabove x
proof
  let L be non empty reflexive transitive RelStr, x,y be Element of L;
  assume
A1: x <= y;
  hereby
    let z be object;
    assume z in waybelow x;
    then consider v being Element of L such that
A2: z = v and
A3: v << x;
    v << y by A1,A3,Th2;
    hence z in waybelow y by A2;
  end;
  let z be object;
  assume z in wayabove y;
  then consider v being Element of L such that
A4: z = v and
A5: v >> y;
  v >> x by A1,A5,Th2;
  hence thesis by A4;
end;
