
theorem Th5:
  for L being RelStr holds L is antisymmetric iff L opp is antisymmetric
proof
  let L be RelStr;
  thus L is antisymmetric implies L opp is antisymmetric
  proof
    assume
A1: L is antisymmetric;
    let x,y be Element of L opp;
    assume x <= y & x >= y;
    then ~x >= ~y & ~x <= ~y by Th1;
    hence thesis by A1;
  end;
  assume
A2: L opp is antisymmetric;
  let x,y be Element of L;
  assume x <= y & x >= y;
  then x~ >= y~ & x~ <= y~ by LATTICE3:9;
  hence thesis by A2;
end;
