
theorem Th17:
  for L being non empty RelStr holds L is complete iff L opp is complete
proof
  let L be non empty RelStr;
A1: now
    let L be non empty RelStr;
    assume
A2: L is complete;
    thus L opp is complete
    proof
      let X be set;
      set Y = {x where x is Element of L: x is_<=_than X};
      consider a being Element of L such that
A3:   Y is_<=_than a and
A4:   for b being Element of L st Y is_<=_than b holds a <= b by A2;
      take x = a~;
      thus X is_<=_than x
      proof
        let y be Element of L opp;
        assume
A5:     y in X;
        Y is_<=_than ~y
        proof
          let b be Element of L;
          assume b in Y;
          then ex z being Element of L st b = z & z is_<=_than X;
          hence thesis by A5;
        end;
        hence thesis by Th2,A4;
      end;
      let y be Element of L opp;
      assume X is_<=_than y;
      then X is_>=_than ~y by Th9;
      then ~y in Y;
      then
A6:   a >= ~y by A3;
      ~x = x;
      hence thesis by A6,Th1;
    end;
  end;
  L opp~ is complete implies L is complete by YELLOW_0:3;
  hence thesis by A1;
end;
